From 27434a86c841365a59d27e44cc480bd6bcfb0b68 Mon Sep 17 00:00:00 2001 From: Ian Wright <55286208+Z80coder@users.noreply.github.com> Date: Tue, 31 Jan 2023 09:15:52 +0000 Subject: [PATCH 001/113] iris data --- tests/data/iris.data | 150 +++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 150 insertions(+) create mode 100644 tests/data/iris.data diff --git a/tests/data/iris.data b/tests/data/iris.data new file mode 100644 index 0000000..a3490e0 --- /dev/null +++ b/tests/data/iris.data @@ -0,0 +1,150 @@ +5.1,3.5,1.4,0.2,Iris-setosa +4.9,3.0,1.4,0.2,Iris-setosa +4.7,3.2,1.3,0.2,Iris-setosa +4.6,3.1,1.5,0.2,Iris-setosa +5.0,3.6,1.4,0.2,Iris-setosa +5.4,3.9,1.7,0.4,Iris-setosa +4.6,3.4,1.4,0.3,Iris-setosa +5.0,3.4,1.5,0.2,Iris-setosa +4.4,2.9,1.4,0.2,Iris-setosa +4.9,3.1,1.5,0.1,Iris-setosa +5.4,3.7,1.5,0.2,Iris-setosa +4.8,3.4,1.6,0.2,Iris-setosa +4.8,3.0,1.4,0.1,Iris-setosa +4.3,3.0,1.1,0.1,Iris-setosa +5.8,4.0,1.2,0.2,Iris-setosa +5.7,4.4,1.5,0.4,Iris-setosa 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1 1 0 0 0 1 2 +0 1 0 0 0 0 0 1 0 0 1 1 0 0 0 1 2 +0 0 1 1 0 0 0 1 0 0 1 1 0 0 0 1 2 +0 1 0 0 0 0 0 1 0 0 1 1 0 0 0 1 2 +0 0 1 1 0 0 1 0 0 0 1 1 0 0 0 1 2 +0 0 1 1 0 0 0 1 0 0 1 1 0 0 0 1 2 \ No newline at end of file diff --git a/tests/test_iris.py b/tests/test_iris.py new file mode 100644 index 0000000..375d0ae --- /dev/null +++ b/tests/test_iris.py @@ -0,0 +1,302 @@ +from pathlib import Path + +import jax +import ml_collections +import numpy +import optax +from flax.training import train_state +from tqdm import tqdm + +from neurallogic import ( + hard_and, + hard_majority, + hard_not, + hard_or, + hard_xor, + hard_dropout, + hard_masks, + real_encoder, + harden, + harden_layer, + neural_logic_net, +) +from tests import utils + + +def check_symbolic(nets, data, trained_state): + x_training, y_training, x_test, y_test = get_train_and_test_data(data) + _, hard, symbolic = nets + _, test_loss, test_accuracy = apply_model_with_grad(trained_state, x_test, y_test) + print( + "soft_net: final test_loss: %.4f, final test_accuracy: %.2f" + % (test_loss, test_accuracy * 100) + ) + hard_weights = harden.hard_weights(trained_state.params) + hard_trained_state = train_state.TrainState.create( + apply_fn=hard.apply, params=hard_weights, tx=optax.sgd(1.0, 1.0) + ) + hard_input = harden.harden(x_test) + hard_test_accuracy = apply_hard_model(hard_trained_state, hard_input, y_test) + print("hard_net: final test_accuracy: %.2f" % (hard_test_accuracy * 100)) + assert numpy.isclose(test_accuracy, hard_test_accuracy, atol=0.0001) + if True: + symbolic_weights = hard_weights # utils.make_symbolic(hard_weights) + symbolic_trained_state = train_state.TrainState.create( + apply_fn=symbolic.apply, params=symbolic_weights, tx=optax.sgd(1.0, 1.0) + ) + symbolic_input = hard_input.tolist() + symbolic_test_accuracy = apply_hard_model( + symbolic_trained_state, symbolic_input, y_test + ) + print( + "symbolic_net: final test_accuracy: %.2f" % (symbolic_test_accuracy * 100) + ) + assert numpy.isclose(test_accuracy, symbolic_test_accuracy, atol=0.0001) + if False: + # CPU and GPU give different results, so we can't easily regress on a static symbolic expression + symbolic_input = [f"x{i}" for i in range(len(hard_input[0].tolist()))] + symbolic_output = symbolic.apply({"params": symbolic_weights}, symbolic_input) + print("symbolic_output", symbolic_output[0][:10000]) + + +num_features = 4 +num_classes = 3 + + +def get_data(): + data_dir = Path(__file__).parent.parent / "tests" / "data" + data = numpy.loadtxt( + data_dir / "iris.data", + delimiter=",", + dtype={ + "names": ( + "sepal_length", + "sepal_width", + "petal_length", + "petal_width", + "class", + ), + "formats": ("f4", "f4", "f4", "f4", "U15"), + }, + ) + features = numpy.array([list(d)[:4] for d in data]) + # Normalise each feature column to be in the range [0, 1] + features = (features - features.min(axis=0)) / ( + features.max(axis=0) - features.min(axis=0) + ) + labels = numpy.array( + [ + 0 + if d[num_features] == "Iris-setosa" + else 1 + if d[num_features] == "Iris-versicolor" + else 2 + for d in data + ] + ) + return features, labels + + +# overfitting model: 100% training accuracy +def nln(type, x, training: bool): + input_size = x.shape[0] + bits_per_feature = 10 + x = real_encoder.real_encoder_layer(type)(bits_per_feature)(x) + x = x.ravel() + dtype = jax.numpy.float32 + mask_layer_size = 120 + x = hard_masks.mask_to_true_layer(type)(mask_layer_size, dtype=dtype)(x) + x = x.reshape((mask_layer_size, input_size * bits_per_feature)) + x = hard_majority.majority_layer(type)()(x) + x = hard_not.not_layer(type)(18)(x) + x = x.ravel() + ######################################################## + x = harden_layer.harden_layer(type)(x) + x = x.reshape((num_classes, int(x.shape[0] / num_classes))) + x = x.sum(-1) + return x + + +def batch_nln(type, x, training: bool): + return jax.vmap(lambda x: nln(type, x, training))(x) + + +class TrainState(train_state.TrainState): + dropout_rng: jax.random.KeyArray + + +def create_train_state(net, rng, dropout_rng, config): + mock_input = jax.numpy.ones([1, num_features]) + soft_weights = net.init(rng, mock_input, training=False)["params"] + tx = optax.sgd(config.learning_rate, config.momentum) + return TrainState.create( + apply_fn=net.apply, params=soft_weights, tx=tx, dropout_rng=dropout_rng + ) + + +@jax.jit +def update_model(state, grads): + return state.apply_gradients(grads=grads) + + +def apply_model_with_grad_impl(state, features, labels, dropout_rng, training: bool): + dropout_train_rng = jax.random.fold_in(key=dropout_rng, data=state.step) + + def loss_fn(params): + logits = state.apply_fn( + {"params": params}, + features, + training=training, + rngs={"dropout": dropout_train_rng}, + ) + one_hot = jax.nn.one_hot(labels, num_classes) + loss = jax.numpy.mean( + optax.softmax_cross_entropy(logits=logits, labels=one_hot) + ) + return loss, logits + + grad_fn = jax.value_and_grad(loss_fn, has_aux=True) + (loss, logits), grads = grad_fn(state.params) + accuracy = jax.numpy.mean(jax.numpy.argmax(logits, -1) == labels) + return grads, loss, accuracy + + +@jax.jit +def apply_model_with_grad_and_training(state, features, labels, dropout_rng): + return apply_model_with_grad_impl( + state, features, labels, dropout_rng, training=True + ) + + +@jax.jit +def apply_model_with_grad(state, features, labels, dropout_rng): + return apply_model_with_grad_impl( + state, features, labels, dropout_rng, training=False + ) + + +def train_epoch(state, features, labels, batch_size, rng, dropout_rng): + train_ds_size = len(features) + steps_per_epoch = train_ds_size // batch_size + + perms = jax.random.permutation(rng, len(features)) + perms = perms[: steps_per_epoch * batch_size] # skip incomplete batch + perms = perms.reshape((steps_per_epoch, batch_size)) + + epoch_loss = [] + epoch_accuracy = [] + + for perm in perms: + batch_features = features[perm, ...] + batch_labels = labels[perm, ...] + grads, loss, accuracy = apply_model_with_grad_and_training( + state, batch_features, batch_labels, dropout_rng + ) + state = update_model(state, grads) + epoch_loss.append(loss) + epoch_accuracy.append(accuracy) + train_loss = numpy.mean(epoch_loss) + train_accuracy = numpy.mean(epoch_accuracy) + return state, train_loss, train_accuracy + + +def train_and_evaluate( + init_rng, dropout_rng, net, data, config: ml_collections.ConfigDict +): + state = create_train_state(net, init_rng, dropout_rng, config) + x_training, y_training, x_test, y_test = data + best_train_accuracy = 0.0 + best_test_accuracy = 0.0 + for epoch in range(1, config.num_epochs + 1): + init_rng, input_rng = jax.random.split(init_rng) + state, train_loss, train_accuracy = train_epoch( + state, x_training, y_training, config.batch_size, input_rng, dropout_rng + ) + _, test_loss, test_accuracy = apply_model_with_grad( + state, x_test, y_test, dropout_rng + ) + if train_accuracy > best_train_accuracy: + best_train_accuracy = train_accuracy + # print(f"best_train_accuracy: {best_train_accuracy * 100:.2f}") + if test_accuracy >= best_test_accuracy: + best_test_accuracy = test_accuracy + # print(f"best_test_accuracy: {best_test_accuracy * 100:.2f}") + # else: + # print(f"test_accuracy: {test_accuracy * 100:.2f}") + # print("\n") + + # print( + # "epoch:% 3d, train_loss: %.4f, train_accuracy: %.2f, test_loss: %.4f, test_accuracy: %.2f" + # % (epoch, train_loss, train_accuracy * 100, test_loss, test_accuracy * 100) + # ) + + return state + + +def apply_hard_model(state, features, label): + def logits_fn(params): + return state.apply_fn({"params": params}, features, training=False) + + logits = logits_fn(state.params) + if isinstance(logits, list): + logits = jax.numpy.array(logits) + logits *= 1.0 + accuracy = jax.numpy.mean(jax.numpy.argmax(logits, -1) == label) + return accuracy + + +def apply_hard_model_to_data(state, features, labels): + accuracy = 0 + for (image, label) in tqdm(zip(features, labels), total=len(features)): + accuracy += apply_hard_model(state, image, label) + return accuracy / len(features) + + +def get_config(): + config = ml_collections.ConfigDict() + config.learning_rate = 0.01 + config.momentum = 0.9 + config.batch_size = 120 + config.num_epochs = 2 + return config + + +def train_test_split(features, labels, rng, test_size=0.2): + rng, split_rng = jax.random.split(rng) + train_size = int(len(features) * (1 - test_size)) + train_idx = jax.random.permutation(split_rng, len(features))[:train_size] + test_idx = jax.random.permutation(split_rng, len(features))[train_size:] + return ( + features[train_idx], + features[test_idx], + labels[train_idx], + labels[test_idx], + ) + + +def test_iris(): + rng = jax.random.PRNGKey(0) + rng, int_rng, dropout_rng = jax.random.split(rng, 3) + # Train net + soft, hard, symbolic = neural_logic_net.net( + lambda type, x, training: batch_nln(type, x, training) + ) + features, labels = get_data() + print(soft.tabulate(rng, features[0:1], training=False)) + + # Split features and labels into 80% training and 20% test + x_training, x_test, y_training, y_test = train_test_split( + features, labels, rng, test_size=0.2 + ) + + trained_state = train_and_evaluate( + int_rng, + dropout_rng, + soft, + (x_training, y_training, x_test, y_test), + get_config(), + ) + + # Check symbolic net + # _, hard, symbolic = neural_logic_net.net(lambda type, x: nln(type, x)) + # check_symbolic((soft, hard, symbolic), (training_data, test_data), trained_state) From 6c271d5579e60499cb995673f42ba4d5c555492e Mon Sep 17 00:00:00 2001 From: Ian Wright <55286208+Z80coder@users.noreply.github.com> Date: Tue, 31 Jan 2023 10:53:12 +0000 Subject: [PATCH 003/113] fix linter problem --- tests/test_iris.py | 14 +++++++------- 1 file changed, 7 insertions(+), 7 deletions(-) diff --git a/tests/test_iris.py b/tests/test_iris.py index 375d0ae..e804ed6 100644 --- a/tests/test_iris.py +++ b/tests/test_iris.py @@ -24,7 +24,7 @@ def check_symbolic(nets, data, trained_state): - x_training, y_training, x_test, y_test = get_train_and_test_data(data) + x_training, y_training, x_test, y_test = data _, hard, symbolic = nets _, test_loss, test_accuracy = apply_model_with_grad(trained_state, x_test, y_test) print( @@ -217,13 +217,13 @@ def train_and_evaluate( ) if train_accuracy > best_train_accuracy: best_train_accuracy = train_accuracy - # print(f"best_train_accuracy: {best_train_accuracy * 100:.2f}") + print(f"best_train_accuracy: {best_train_accuracy * 100:.2f}") if test_accuracy >= best_test_accuracy: best_test_accuracy = test_accuracy - # print(f"best_test_accuracy: {best_test_accuracy * 100:.2f}") - # else: - # print(f"test_accuracy: {test_accuracy * 100:.2f}") - # print("\n") + print(f"best_test_accuracy: {best_test_accuracy * 100:.2f}") + else: + print(f"test_accuracy: {test_accuracy * 100:.2f}") + print("\n") # print( # "epoch:% 3d, train_loss: %.4f, train_accuracy: %.2f, test_loss: %.4f, test_accuracy: %.2f" @@ -299,4 +299,4 @@ def test_iris(): # Check symbolic net # _, hard, symbolic = neural_logic_net.net(lambda type, x: nln(type, x)) - # check_symbolic((soft, hard, symbolic), (training_data, test_data), trained_state) + # check_symbolic((soft, hard, symbolic), (x_training, y_training, x_test, y_test), trained_state) From 4d6055ebe1a38b3701b4d0f73312ce2413a79558 Mon Sep 17 00:00:00 2001 From: Ian Wright <55286208+Z80coder@users.noreply.github.com> Date: Wed, 1 Feb 2023 10:05:33 +0000 Subject: [PATCH 004/113] learn on binary iris data --- tests/test_iris.py | 63 +++++++++++++++++++++++++++++++++++----------- 1 file changed, 49 insertions(+), 14 deletions(-) diff --git a/tests/test_iris.py b/tests/test_iris.py index e804ed6..113c087 100644 --- a/tests/test_iris.py +++ b/tests/test_iris.py @@ -59,11 +59,12 @@ def check_symbolic(nets, data, trained_state): print("symbolic_output", symbolic_output[0][:10000]) -num_features = 4 +binary_iris = True +num_features = 16 if binary_iris else 4 num_classes = 3 -def get_data(): +def get_iris_data(): data_dir = Path(__file__).parent.parent / "tests" / "data" data = numpy.loadtxt( data_dir / "iris.data", @@ -97,8 +98,16 @@ def get_data(): return features, labels +def get_binary_iris_data(): + data_dir = Path(__file__).parent.parent / "tests" / "data" + data = numpy.loadtxt(data_dir / "BinaryIrisData.txt").astype(dtype=numpy.int32) + features = data[:, 0:num_features] # Input features + labels = data[:, num_features] # Target value + return features, labels + + # overfitting model: 100% training accuracy -def nln(type, x, training: bool): +def nln_iris(type, x, training: bool): input_size = x.shape[0] bits_per_feature = 10 x = real_encoder.real_encoder_layer(type)(bits_per_feature)(x) @@ -117,8 +126,26 @@ def nln(type, x, training: bool): return x -def batch_nln(type, x, training: bool): - return jax.vmap(lambda x: nln(type, x, training))(x) +def nln_binary_iris(type, x, training: bool): + dtype = jax.numpy.float32 + mask_layer_size = 100 + x = hard_masks.mask_to_true_layer(type)(mask_layer_size, dtype=dtype)(x) + x = hard_majority.majority_layer(type)()(x) + x = hard_not.not_layer(type)(9)(x) + x = x.ravel() + ######################################################## + x = harden_layer.harden_layer(type)(x) + x = x.reshape((num_classes, int(x.shape[0] / num_classes))) + x = x.sum(-1) + return x + + +def batch_nln_iris(type, x, training: bool): + return jax.vmap(lambda x: nln_iris(type, x, training))(x) + + +def batch_nln_binary_iris(type, x, training: bool): + return jax.vmap(lambda x: nln_binary_iris(type, x, training))(x) class TrainState(train_state.TrainState): @@ -217,13 +244,14 @@ def train_and_evaluate( ) if train_accuracy > best_train_accuracy: best_train_accuracy = train_accuracy - print(f"best_train_accuracy: {best_train_accuracy * 100:.2f}") + print(f"eopch: {epoch}") + print(f"\tbest_train_accuracy: {best_train_accuracy * 100:.2f}") if test_accuracy >= best_test_accuracy: best_test_accuracy = test_accuracy - print(f"best_test_accuracy: {best_test_accuracy * 100:.2f}") + print(f"\tbest_test_accuracy: {best_test_accuracy * 100:.2f}") else: - print(f"test_accuracy: {test_accuracy * 100:.2f}") - print("\n") + print(f"\ttest_accuracy: {test_accuracy * 100:.2f}") + print("\n") # print( # "epoch:% 3d, train_loss: %.4f, train_accuracy: %.2f, test_loss: %.4f, test_accuracy: %.2f" @@ -257,7 +285,7 @@ def get_config(): config.learning_rate = 0.01 config.momentum = 0.9 config.batch_size = 120 - config.num_epochs = 2 + config.num_epochs = 2 # 400 return config @@ -278,10 +306,17 @@ def test_iris(): rng = jax.random.PRNGKey(0) rng, int_rng, dropout_rng = jax.random.split(rng, 3) # Train net - soft, hard, symbolic = neural_logic_net.net( - lambda type, x, training: batch_nln(type, x, training) - ) - features, labels = get_data() + if binary_iris: + features, labels = get_binary_iris_data() + soft, hard, symbolic = neural_logic_net.net( + lambda type, x, training: batch_nln_binary_iris(type, x, training) + ) + else: + features, labels = get_iris_data() + soft, hard, symbolic = neural_logic_net.net( + lambda type, x, training: batch_nln_iris(type, x, training) + ) + print(soft.tabulate(rng, features[0:1], training=False)) # Split features and labels into 80% training and 20% test From 2c3688b8149b4632ad94b778839daa0f5dd25db1 Mon Sep 17 00:00:00 2001 From: Ian Wright <55286208+Z80coder@users.noreply.github.com> Date: Wed, 1 Feb 2023 12:37:48 +0000 Subject: [PATCH 005/113] collect stats --- tests/test_iris.py | 105 +++++++++++++++++++++++++++++++++++---------- 1 file changed, 82 insertions(+), 23 deletions(-) diff --git a/tests/test_iris.py b/tests/test_iris.py index 113c087..108a061 100644 --- a/tests/test_iris.py +++ b/tests/test_iris.py @@ -126,12 +126,37 @@ def nln_iris(type, x, training: bool): return x -def nln_binary_iris(type, x, training: bool): +def nln_binary_iris_95_27(type, x, training: bool): + dtype = jax.numpy.float32 + mask_layer_size = 90 + x = hard_masks.mask_to_true_layer(type)(mask_layer_size, dtype=dtype)(x) + x = hard_majority.majority_layer(type)()(x) + x = x.ravel() + ######################################################## + x = harden_layer.harden_layer(type)(x) + x = x.reshape((num_classes, int(x.shape[0] / num_classes))) + x = x.sum(-1) + return x + + +def nln_binary_iris_95_6(type, x, training: bool): + dtype = jax.numpy.float32 + mask_layer_size = 120 + x = hard_masks.mask_to_true_layer(type)(mask_layer_size, dtype=dtype)(x) + x = hard_majority.majority_layer(type)()(x) + x = x.ravel() + ######################################################## + x = harden_layer.harden_layer(type)(x) + x = x.reshape((num_classes, int(x.shape[0] / num_classes))) + x = x.sum(-1) + return x + + +def nln_binary_iris_95_53(type, x, training: bool): dtype = jax.numpy.float32 - mask_layer_size = 100 + mask_layer_size = 150 x = hard_masks.mask_to_true_layer(type)(mask_layer_size, dtype=dtype)(x) x = hard_majority.majority_layer(type)()(x) - x = hard_not.not_layer(type)(9)(x) x = x.ravel() ######################################################## x = harden_layer.harden_layer(type)(x) @@ -140,6 +165,30 @@ def nln_binary_iris(type, x, training: bool): return x +def nln_binary_iris_95_7(type, x, training: bool): + dtype = jax.numpy.float32 + mask_layer_size = 900 + x = hard_masks.mask_to_true_layer(type)(mask_layer_size, dtype=dtype)(x) + x = hard_majority.majority_layer(type)()(x) + ######################################################## + x = harden_layer.harden_layer(type)(x) + x = x.reshape((num_classes, int(x.shape[0] / num_classes))) + x = x.sum(-1) + return x + + +def nln_binary_iris(type, x, training: bool): + dtype = jax.numpy.float32 + mask_layer_size = 120 + x = hard_masks.mask_to_true_layer(type)(mask_layer_size, dtype=dtype)(x) + x = hard_majority.majority_layer(type)()(x) + ######################################################## + x = harden_layer.harden_layer(type)(x) + x = x.reshape((num_classes, int(x.shape[0] / num_classes))) + x = x.sum(-1) + return x + + def batch_nln_iris(type, x, training: bool): return jax.vmap(lambda x: nln_iris(type, x, training))(x) @@ -244,21 +293,22 @@ def train_and_evaluate( ) if train_accuracy > best_train_accuracy: best_train_accuracy = train_accuracy - print(f"eopch: {epoch}") - print(f"\tbest_train_accuracy: {best_train_accuracy * 100:.2f}") + # print(f"epoch: {epoch}") + # print(f"\tbest_train_accuracy: {best_train_accuracy * 100:.2f}") if test_accuracy >= best_test_accuracy: best_test_accuracy = test_accuracy - print(f"\tbest_test_accuracy: {best_test_accuracy * 100:.2f}") - else: - print(f"\ttest_accuracy: {test_accuracy * 100:.2f}") - print("\n") + # print(f"\tbest_test_accuracy: {best_test_accuracy * 100:.2f}") + # else: + # print(f"\ttest_accuracy: {test_accuracy * 100:.2f}") + # print("\n") # print( # "epoch:% 3d, train_loss: %.4f, train_accuracy: %.2f, test_loss: %.4f, test_accuracy: %.2f" # % (epoch, train_loss, train_accuracy * 100, test_loss, test_accuracy * 100) # ) - return state + # return trained state and final test_accuracy + return state, test_accuracy def apply_hard_model(state, features, label): @@ -285,7 +335,7 @@ def get_config(): config.learning_rate = 0.01 config.momentum = 0.9 config.batch_size = 120 - config.num_epochs = 2 # 400 + config.num_epochs = 400 return config @@ -303,8 +353,6 @@ def train_test_split(features, labels, rng, test_size=0.2): def test_iris(): - rng = jax.random.PRNGKey(0) - rng, int_rng, dropout_rng = jax.random.split(rng, 3) # Train net if binary_iris: features, labels = get_binary_iris_data() @@ -317,19 +365,30 @@ def test_iris(): lambda type, x, training: batch_nln_iris(type, x, training) ) + rng = jax.random.PRNGKey(0) print(soft.tabulate(rng, features[0:1], training=False)) - # Split features and labels into 80% training and 20% test - x_training, x_test, y_training, y_test = train_test_split( - features, labels, rng, test_size=0.2 - ) + num_experiments = 25 + final_test_accuracies = [] + for i in range(num_experiments): + # Split features and labels into 80% training and 20% test + rng, int_rng, dropout_rng = jax.random.split(rng, 3) + x_training, x_test, y_training, y_test = train_test_split( + features, labels, rng, test_size=0.2 + ) + trained_state, final_test_accuracy = train_and_evaluate( + int_rng, + dropout_rng, + soft, + (x_training, y_training, x_test, y_test), + get_config(), + ) + final_test_accuracies.append(final_test_accuracy) + print(f"{i}: final test accuracy: {final_test_accuracy * 100:.2f}") - trained_state = train_and_evaluate( - int_rng, - dropout_rng, - soft, - (x_training, y_training, x_test, y_test), - get_config(), + # print mean, min, max, lowest 5%, highest 5% of final test accuracies + print( + f"mean: {numpy.mean(final_test_accuracies) * 100:.2f}, min: {numpy.min(final_test_accuracies) * 100:.2f}, max: {numpy.max(final_test_accuracies) * 100:.2f}, lowest 5%: {numpy.quantile(final_test_accuracies, 0.05) * 100:.2f}, highest 5%: {numpy.quantile(final_test_accuracies, 0.95) * 100:.2f}" ) # Check symbolic net From 4e2d0dcda6d721bfc190e26f1fbff2856acc0a0a Mon Sep 17 00:00:00 2001 From: Ian Wright <55286208+Z80coder@users.noreply.github.com> Date: Thu, 2 Feb 2023 13:15:53 +0000 Subject: [PATCH 006/113] *** reproducible 2nd place on Iris leaderboard *** --- tests/test_iris.py | 102 ++++++++++++++++----------------------------- 1 file changed, 37 insertions(+), 65 deletions(-) diff --git a/tests/test_iris.py b/tests/test_iris.py index 108a061..64d6b39 100644 --- a/tests/test_iris.py +++ b/tests/test_iris.py @@ -4,21 +4,22 @@ import ml_collections import numpy import optax +import scipy from flax.training import train_state from tqdm import tqdm from neurallogic import ( hard_and, + hard_dropout, hard_majority, + hard_masks, hard_not, hard_or, hard_xor, - hard_dropout, - hard_masks, - real_encoder, harden, harden_layer, neural_logic_net, + real_encoder, ) from tests import utils @@ -126,62 +127,29 @@ def nln_iris(type, x, training: bool): return x -def nln_binary_iris_95_27(type, x, training: bool): - dtype = jax.numpy.float32 - mask_layer_size = 90 - x = hard_masks.mask_to_true_layer(type)(mask_layer_size, dtype=dtype)(x) - x = hard_majority.majority_layer(type)()(x) - x = x.ravel() - ######################################################## - x = harden_layer.harden_layer(type)(x) - x = x.reshape((num_classes, int(x.shape[0] / num_classes))) - x = x.sum(-1) - return x - - -def nln_binary_iris_95_6(type, x, training: bool): - dtype = jax.numpy.float32 - mask_layer_size = 120 - x = hard_masks.mask_to_true_layer(type)(mask_layer_size, dtype=dtype)(x) - x = hard_majority.majority_layer(type)()(x) - x = x.ravel() - ######################################################## - x = harden_layer.harden_layer(type)(x) - x = x.reshape((num_classes, int(x.shape[0] / num_classes))) - x = x.sum(-1) - return x - - -def nln_binary_iris_95_53(type, x, training: bool): - dtype = jax.numpy.float32 - mask_layer_size = 150 - x = hard_masks.mask_to_true_layer(type)(mask_layer_size, dtype=dtype)(x) - x = hard_majority.majority_layer(type)()(x) - x = x.ravel() - ######################################################## - x = harden_layer.harden_layer(type)(x) - x = x.reshape((num_classes, int(x.shape[0] / num_classes))) - x = x.sum(-1) - return x - - -def nln_binary_iris_95_7(type, x, training: bool): - dtype = jax.numpy.float32 - mask_layer_size = 900 - x = hard_masks.mask_to_true_layer(type)(mask_layer_size, dtype=dtype)(x) - x = hard_majority.majority_layer(type)()(x) - ######################################################## - x = harden_layer.harden_layer(type)(x) - x = x.reshape((num_classes, int(x.shape[0] / num_classes))) - x = x.sum(-1) - return x +""" +| Technique/Accuracy | Mean | 5 %ile | 95 %ile | Min | Max | +| ------------------ | -------------- | ------- | ------- | ------ | ------ | +| Tsetlin | 95.0 +/- 0.2 | 86.7 | 100.0 | 80.0 | 100.0 | +| dB | 94.2 +/- 0.1 | 86.7 | 100.0 | 80.0 | 100.0 | +| Neural network | 93.8 +/- 0.2 | 86.7 | 100.0 | 80.0 | 100.0 | +| SVM | 93.6 +/- 0.3 | 86.7 | 100.0 | 76.7 | 100.0 | +| Naive Bayes | 91.6 +/- 0.3 | 83.3 | 96.7 | 70.0 | 100.0 | +Source: https://arxiv.org/pdf/1804.01508.pdf +""" +# mean: 94.18, sem: 0.13, min: 80.00, max: 100.00, 5%: 86.67, 95%: 100.00 def nln_binary_iris(type, x, training: bool): dtype = jax.numpy.float32 - mask_layer_size = 120 - x = hard_masks.mask_to_true_layer(type)(mask_layer_size, dtype=dtype)(x) + x = hard_masks.mask_to_true_layer(type)(120, dtype=dtype)(x) x = hard_majority.majority_layer(type)()(x) + x = hard_dropout.hard_dropout(type)( + rate=0.25, + dropout_value=0.0, + deterministic=not training, + dtype=dtype, + )(x) ######################################################## x = harden_layer.harden_layer(type)(x) x = x.reshape((num_classes, int(x.shape[0] / num_classes))) @@ -302,10 +270,10 @@ def train_and_evaluate( # print(f"\ttest_accuracy: {test_accuracy * 100:.2f}") # print("\n") - # print( - # "epoch:% 3d, train_loss: %.4f, train_accuracy: %.2f, test_loss: %.4f, test_accuracy: %.2f" - # % (epoch, train_loss, train_accuracy * 100, test_loss, test_accuracy * 100) - # ) + print( + "epoch:% 3d, train_loss: %.4f, train_accuracy: %.2f, test_loss: %.4f, test_accuracy: %.2f" + % (epoch, train_loss, train_accuracy * 100, test_loss, test_accuracy * 100) + ) # return trained state and final test_accuracy return state, test_accuracy @@ -335,7 +303,7 @@ def get_config(): config.learning_rate = 0.01 config.momentum = 0.9 config.batch_size = 120 - config.num_epochs = 400 + config.num_epochs = 2 # 500 for paper return config @@ -368,7 +336,7 @@ def test_iris(): rng = jax.random.PRNGKey(0) print(soft.tabulate(rng, features[0:1], training=False)) - num_experiments = 25 + num_experiments = 1 # 1000 for paper final_test_accuracies = [] for i in range(num_experiments): # Split features and labels into 80% training and 20% test @@ -385,11 +353,15 @@ def test_iris(): ) final_test_accuracies.append(final_test_accuracy) print(f"{i}: final test accuracy: {final_test_accuracy * 100:.2f}") - - # print mean, min, max, lowest 5%, highest 5% of final test accuracies - print( - f"mean: {numpy.mean(final_test_accuracies) * 100:.2f}, min: {numpy.min(final_test_accuracies) * 100:.2f}, max: {numpy.max(final_test_accuracies) * 100:.2f}, lowest 5%: {numpy.quantile(final_test_accuracies, 0.05) * 100:.2f}, highest 5%: {numpy.quantile(final_test_accuracies, 0.95) * 100:.2f}" - ) + # print mean, standard error of the mean, min, max, lowest 5%, highest 5% of final test accuracies + print( + f"mean: {numpy.mean(final_test_accuracies) * 100:.2f}, " + f"sem: {scipy.stats.sem(final_test_accuracies) * 100:.2f}, " + f"min: {numpy.min(final_test_accuracies) * 100:.2f}, " + f"max: {numpy.max(final_test_accuracies) * 100:.2f}, " + f"5%: {numpy.percentile(final_test_accuracies, 5) * 100:.2f}, " + f"95%: {numpy.percentile(final_test_accuracies, 95) * 100:.2f}" + ) # Check symbolic net # _, hard, symbolic = neural_logic_net.net(lambda type, x: nln(type, x)) From ee32063b8294a07b14b3b26c8300c640b819baa5 Mon Sep 17 00:00:00 2001 From: Ian Wright <55286208+Z80coder@users.noreply.github.com> Date: Mon, 6 Feb 2023 18:05:31 +0000 Subject: [PATCH 007/113] *** 95.87% on noisy xor bookmark *** --- neurallogic/hard_and.py | 5 +- neurallogic/harden_layer.py | 20 +- neurallogic/neural_logic_net.py | 8 +- tests/test_iris.py | 1 + tests/test_noisy_xor.py | 336 +++++++++++++++++++++++++++----- 5 files changed, 311 insertions(+), 59 deletions(-) diff --git a/neurallogic/hard_and.py b/neurallogic/hard_and.py index 7aa6867..2fd58b1 100644 --- a/neurallogic/hard_and.py +++ b/neurallogic/hard_and.py @@ -23,14 +23,15 @@ def hard_and_neuron(w, x): # TODO: move initialization to separate file -def initialize_near_to_zero(): +# TODO: simplify initialization to avoid the need to specify a guassian mean and std +def initialize_near_to_zero(mean=-1, std=0.5): # TODO: investigate better initialization def init(key, shape, dtype): dtype = jax.dtypes.canonicalize_dtype(dtype) # Sample from standard normal distribution (zero mean, unit variance) x = jax.random.normal(key, shape, dtype) # Transform to a normal distribution with mean -1 and standard deviation 0.5 - x = 0.5 * x - 1 + x = std * x + mean x = jax.numpy.clip(x, 0.001, 0.999) return x diff --git a/neurallogic/harden_layer.py b/neurallogic/harden_layer.py index a2d6665..0024a98 100644 --- a/neurallogic/harden_layer.py +++ b/neurallogic/harden_layer.py @@ -7,21 +7,27 @@ def harden_element(x): # non-differentiable return jax.lax.cond(x > 0.5, lambda _: 1.0, lambda _: 0.0, None) + def straight_through_harden_element(x): - # Create an exactly-zero expression with Sterbenz lemma that has - # an exactly-one gradient. - zero = x - jax.lax.stop_gradient(x) - return zero + jax.lax.stop_gradient(harden_element(x)) + # Create an exactly-zero expression with Sterbenz lemma that has + # an exactly-one gradient. + zero = x - jax.lax.stop_gradient(x) + return zero + jax.lax.stop_gradient(harden_element(x)) + soft_harden_layer = jax.vmap(straight_through_harden_element) + def hard_harden_layer(x): return x -#TODO: can we harden arbitrary tensors? -#TODO: is this correct? + +# TODO: can we harden arbitrary tensors? +# TODO: is this correct? def symbolic_harden_layer(x): return x -harden_layer = neural_logic_net.select(soft_harden_layer, hard_harden_layer, symbolic_harden_layer) +harden_layer = neural_logic_net.select( + soft_harden_layer, hard_harden_layer, symbolic_harden_layer +) diff --git a/neurallogic/neural_logic_net.py b/neurallogic/neural_logic_net.py index a2375ae..4f65489 100644 --- a/neurallogic/neural_logic_net.py +++ b/neurallogic/neural_logic_net.py @@ -21,12 +21,12 @@ def __call__(self, x, **kwargs): class HardNet(nn.Module): @nn.compact - def __call__(self, x): - return f(NetType.Hard, x) + def __call__(self, x, **kwargs): + return f(NetType.Hard, x, **kwargs) class SymbolicNet(nn.Module): @nn.compact - def __call__(self, x): - return f(NetType.Symbolic, x) + def __call__(self, x, **kwargs): + return f(NetType.Symbolic, x, **kwargs) return SoftNet(), HardNet(), SymbolicNet() diff --git a/tests/test_iris.py b/tests/test_iris.py index 64d6b39..1b0b80b 100644 --- a/tests/test_iris.py +++ b/tests/test_iris.py @@ -139,6 +139,7 @@ def nln_iris(type, x, training: bool): Source: https://arxiv.org/pdf/1804.01508.pdf """ + # mean: 94.18, sem: 0.13, min: 80.00, max: 100.00, 5%: 86.67, 95%: 100.00 def nln_binary_iris(type, x, training: bool): dtype = jax.numpy.float32 diff --git a/tests/test_noisy_xor.py b/tests/test_noisy_xor.py index 289302d..759ebff 100644 --- a/tests/test_noisy_xor.py +++ b/tests/test_noisy_xor.py @@ -1,43 +1,57 @@ +import sys from pathlib import Path import jax import ml_collections import numpy import optax +import scipy +from flax import linen as nn from flax.training import train_state +from jax.config import config from tqdm import tqdm from neurallogic import ( hard_and, + hard_dropout, hard_majority, + hard_masks, hard_not, hard_or, hard_xor, - hard_dropout, harden, harden_layer, neural_logic_net, ) from tests import utils +# config.update("jax_enable_x64", True) + -def check_symbolic(nets, data, trained_state): - x_training, y_training, x_test, y_test = get_train_and_test_data(data) +def check_symbolic(nets, data, trained_state, dropout_rng): + x_training, y_training, x_test, y_test = data _, hard, symbolic = nets - _, test_loss, test_accuracy = apply_model_with_grad(trained_state, x_test, y_test) + _, test_loss, test_accuracy = apply_model_with_grad( + trained_state, x_test, y_test, dropout_rng + ) print( "soft_net: final test_loss: %.4f, final test_accuracy: %.2f" % (test_loss, test_accuracy * 100) ) hard_weights = harden.hard_weights(trained_state.params) - hard_trained_state = train_state.TrainState.create( - apply_fn=hard.apply, params=hard_weights, tx=optax.sgd(1.0, 1.0) + hard_trained_state = TrainState.create( + apply_fn=hard.apply, + params=hard_weights, + tx=optax.sgd(1.0, 1.0), + dropout_rng=dropout_rng, ) hard_input = harden.harden(x_test) - hard_test_accuracy = apply_hard_model(hard_trained_state, hard_input, y_test) + hard_test_accuracy = apply_hard_model_to_data( + hard_trained_state, hard_input, y_test + ) print("hard_net: final test_accuracy: %.2f" % (hard_test_accuracy * 100)) assert numpy.isclose(test_accuracy, hard_test_accuracy, atol=0.0001) - if True: + if False: symbolic_weights = hard_weights # utils.make_symbolic(hard_weights) symbolic_trained_state = train_state.TrainState.create( apply_fn=symbolic.apply, params=symbolic_weights, tx=optax.sgd(1.0, 1.0) @@ -75,8 +89,8 @@ def get_data(): return training_data, test_data -# 100% test accuracy -def nln(type, x, training: bool): +# 100% test accuracy (at some point) +def nln_1(type, x, training: bool): x = hard_and.and_layer(type)(20)(x) x = hard_not.not_layer(type)(4)(x) x = x.ravel() @@ -87,17 +101,195 @@ def nln(type, x, training: bool): return x -def nln_experimental(type, x, training: bool): - not_x = jax.numpy.logical_not(x) - input = jax.numpy.concatenate([x, not_x], axis=0) - x = hard_xor.xor_layer(type)(100)(input) - # x = hard_not.not_layer(type)(4)(x) +# 77.83% +def nln_2(type, x, training: bool): + dtype = jax.numpy.float64 + layer_size = 64 # 64 + x = hard_and.and_layer(type)(layer_size)(x) + x = hard_or.or_layer(type)(layer_size)(x) + x = hard_not.not_layer(type)(4)(x) x = x.ravel() - x = hard_dropout.hard_dropout(type)( - rate=0.5, dropout_value=0.0, deterministic=not training - )(x) + x = jax.numpy.array([x]) + x = hard_majority.majority_layer(type)()(x) + z = 1 - x + x = jax.numpy.concatenate([x, z], axis=0) + ######################################################## + # No need to harden here, since we're using majority for 2 classes + x = x.reshape((num_classes, int(x.shape[0] / num_classes))) + x = x.sum(-1) + return x + + +# 81.24 +def nln_3(type, x, training: bool): + y = jax.vmap(lambda x: 1 - x)(x) + x = jax.numpy.concatenate([x, y], axis=0) + dtype = jax.numpy.float64 + layer_size = 32 # 64 + x = hard_and.and_layer(type)(layer_size)(x) + x = hard_or.or_layer(type)(layer_size)(x) + x = hard_not.not_layer(type)(4)(x) + x = x.ravel() + x = jax.numpy.array([x]) + x = hard_majority.majority_layer(type)()(x) + z = 1 - x + x = jax.numpy.concatenate([x, z], axis=0) + ######################################################## + # No need to harden here, since we're using majority for 2 classes + x = x.reshape((num_classes, int(x.shape[0] / num_classes))) + x = x.sum(-1) + return x + + +# 82.30, but lots more high 90s +def nln_4(type, x, training: bool): + y = jax.vmap(lambda x: 1 - x)(x) + x = jax.numpy.concatenate([x, y], axis=0) + dtype = jax.numpy.float64 + layer_size = 64 # 64 + x = hard_and.and_layer(type)(layer_size)(x) + x = hard_or.or_layer(type)(layer_size)(x) + x = hard_not.not_layer(type)(8)(x) + x = x.ravel() + x = x.reshape((64, 8)) + x = hard_majority.majority_layer(type)()(x) + x = x.reshape((8, 8)) + x = hard_majority.majority_layer(type)()(x) + x = x.reshape((1, 8)) + x = hard_majority.majority_layer(type)()(x) + z = jax.vmap(lambda x: 1 - x)(x) + x = jax.numpy.concatenate([x, z], axis=0) + ######################################################## + x = x.reshape((num_classes, int(x.shape[0] / num_classes))) + x = x.sum(-1) + return x + + +# 93.26 +def nln_5(type, x, training: bool): + y = jax.vmap(lambda x: 1 - x)(x) + x = jax.numpy.concatenate([x, y], axis=0) + dtype = jax.numpy.float64 + layer_size = 128 # 64 + x = hard_and.and_layer(type)(layer_size)(x) + x = x.reshape((16, 8)) + x = hard_majority.majority_layer(type)()(x) + x = hard_not.not_layer(type)(8)(x) + x = x.reshape((16, 8)) + x = hard_majority.majority_layer(type)()(x) + x = hard_not.not_layer(type)(2)(x) + x = x.reshape((8, 4)) + x = hard_majority.majority_layer(type)()(x) + x = x.reshape((8, 1)) + x = hard_majority.majority_layer(type)()(x) + z = jax.vmap(lambda x: 1 - x)(x) + x = jax.numpy.concatenate([x, z], axis=0) + ######################################################## + x = x.reshape((num_classes, int(x.shape[0] / num_classes))) + x = x.sum(-1) + return x + + +# 93.90 +def nln_6(type, x, training: bool): + y = jax.vmap(lambda x: 1 - x)(x) + x = jax.numpy.concatenate([x, y], axis=0) + dtype = jax.numpy.float64 + layer_size = 128 + x = hard_and.and_layer(type)(layer_size)(x) + x = hard_not.not_layer(type)(8)(x) + x = x.reshape((64, 16)) + x = hard_majority.majority_layer(type)()(x) + x = hard_not.not_layer(type)(4)(x) + x = x.reshape((16, 16)) + x = hard_majority.majority_layer(type)()(x) + x = hard_not.not_layer(type)(2)(x) + x = x.reshape((8, 4)) + x = hard_majority.majority_layer(type)()(x) + x = x.reshape((8, 1)) + x = hard_majority.majority_layer(type)()(x) + z = jax.vmap(lambda x: 1 - x)(x) + x = jax.numpy.concatenate([x, z], axis=0) + ######################################################## + x = x.reshape((num_classes, int(x.shape[0] / num_classes))) + x = x.sum(-1) + return x + + +# 94.18 with peak_value=0.05 not 0.01 +def nln_7(type, x, training: bool): + y = jax.vmap(lambda x: 1 - x)(x) + x = jax.numpy.concatenate([x, y], axis=0) + dtype = jax.numpy.float64 + layer_size = 256 + x = hard_and.and_layer(type)(layer_size)(x) + x = hard_not.not_layer(type)(4)(x) + x = x.reshape((64, 16)) + x = hard_majority.majority_layer(type)()(x) + x = hard_not.not_layer(type)(4)(x) + x = x.reshape((16, 16)) + x = hard_majority.majority_layer(type)()(x) + x = hard_not.not_layer(type)(2)(x) + x = x.reshape((8, 4)) + x = hard_majority.majority_layer(type)()(x) + x = x.reshape((8, 1)) + x = hard_majority.majority_layer(type)()(x) + z = jax.vmap(lambda x: 1 - x)(x) + x = jax.numpy.concatenate([x, z], axis=0) + ######################################################## + x = x.reshape((num_classes, int(x.shape[0] / num_classes))) + x = x.sum(-1) + return x + + +# mean: 95.00, sem: 0.76, min: 68.84, max: 100.00, 5%: 75.72, 95%: 100.00 +def nln_8(type, x, training: bool): + y = jax.vmap(lambda x: 1 - x)(x) + x = jax.numpy.concatenate([x, y], axis=0) + dtype = jax.numpy.float64 + layer_size = 128 + x = hard_and.and_layer(type)(layer_size)(x) + x = hard_not.not_layer(type)(4)(x) + x = x.reshape((32, 16)) + x = hard_majority.majority_layer(type)()(x) + x = hard_not.not_layer(type)(4)(x) + x = x.reshape((16, 8)) + x = hard_majority.majority_layer(type)()(x) + x = hard_not.not_layer(type)(2)(x) + x = x.reshape((8, 4)) + x = hard_majority.majority_layer(type)()(x) + x = x.reshape((8, 1)) + x = hard_majority.majority_layer(type)()(x) + z = jax.vmap(lambda x: 1 - x)(x) + x = jax.numpy.concatenate([x, z], axis=0) + ######################################################## + x = x.reshape((num_classes, int(x.shape[0] / num_classes))) + x = x.sum(-1) + return x + + +# mean: 95.87, sem: 0.64, min: 64.28, max: 100.00, 5%: 80.49, 95%: 100.00 +def nln(type, x, training: bool): + y = jax.vmap(lambda x: 1 - x)(x) + x = jax.numpy.concatenate([x, y], axis=0) + dtype = jax.numpy.float64 + layer_size = 128 + x = hard_and.and_layer(type)(layer_size)(x) + x = hard_not.not_layer(type)(4)(x) + x = x.reshape((32, 16)) + x = hard_majority.majority_layer(type)()(x) + x = hard_not.not_layer(type)(4)(x) + x = x.reshape((16, 8)) + x = hard_majority.majority_layer(type)()(x) + x = hard_not.not_layer(type)(2)(x) + x = x.reshape((8, 4)) + x = hard_majority.majority_layer(type)()(x) + x = hard_not.not_layer(type)(1)(x) + x = x.reshape((8, 1)) + x = hard_majority.majority_layer(type)()(x) + z = jax.vmap(lambda x: 1 - x)(x) + x = jax.numpy.concatenate([x, z], axis=0) ######################################################## - x = harden_layer.harden_layer(type)(x) x = x.reshape((num_classes, int(x.shape[0] / num_classes))) x = x.sum(-1) return x @@ -114,10 +306,35 @@ class TrainState(train_state.TrainState): def create_train_state(net, rng, dropout_rng, config): mock_input = jax.numpy.ones([1, num_features]) soft_weights = net.init(rng, mock_input, training=False)["params"] - tx = optax.sgd(config.learning_rate, config.momentum) - # tx = optax.yogi(config.learning_rate) - # tx = optax.noisy_sgd(config.learning_rate, config.momentum) - # tx = optax.adagrad(config.learning_rate) # investigate this one + + """ + schedule = optax.linear_onecycle_schedule( + transition_steps=3000, + peak_value=1.0, + pct_start=0.3, + pct_final=0.85, + div_factor=25.0, + final_div_factor=1e4, + ) + schedule = optax.linear_schedule( + init_value=0.01, + end_value=0.01, + transition_steps=config.num_epochs, + transition_begin=0, + ) + """ + schedule = optax.warmup_cosine_decay_schedule( + init_value=0.01, + peak_value=0.01, + warmup_steps=100, + decay_steps=config.num_epochs - 100, + end_value=0.01, + ) + tx = optax.chain( + # optax.clip(1.0), + optax.adamw(learning_rate=schedule), + ) + return TrainState.create( apply_fn=net.apply, params=soft_weights, tx=tx, dropout_rng=dropout_rng ) @@ -223,12 +440,12 @@ def train_and_evaluate( # print(f"test_accuracy: {test_accuracy * 100:.2f}") # print("\n") - # print( - # "epoch:% 3d, train_loss: %.4f, train_accuracy: %.2f, test_loss: %.4f, test_accuracy: %.2f" - # % (epoch, train_loss, train_accuracy * 100, test_loss, test_accuracy * 100) - # ) + print( + "epoch:% 3d, train_loss: %.4f, train_accuracy: %.2f, test_loss: %.4f, test_accuracy: %.2f" + % (epoch, train_loss, train_accuracy * 100, test_loss, test_accuracy * 100) + ) - return state + return state, test_accuracy def apply_hard_model(state, features, label): @@ -254,31 +471,58 @@ def get_config(): config = ml_collections.ConfigDict() config.learning_rate = 0.01 config.momentum = 0.9 - config.batch_size = 256 - config.num_epochs = 500 + config.batch_size = 5000 + config.num_epochs = 800 return config def test_noisy_xor(): - rng = jax.random.PRNGKey(0) - rng, int_rng, dropout_rng = jax.random.split(rng, 3) # Train net - soft, hard, symbolic = neural_logic_net.net( + soft, _, _ = neural_logic_net.net( lambda type, x, training: batch_nln(type, x, training) ) + x_training, y_training, x_test, y_test = get_train_and_test_data(get_data()) - print(soft.tabulate(rng, x_training[0:1], training=False)) - print(f"training_data.shape: {x_training.shape}") - print(f"test_data.shape: {x_test.shape}") - trained_state = train_and_evaluate( - int_rng, - dropout_rng, - soft, - (x_training, y_training, x_test, y_test), - get_config(), - ) + rng = jax.random.PRNGKey(0) + print(soft.tabulate(rng, x_training[0:1], training=False)) - # Check symbolic net - # _, hard, symbolic = neural_logic_net.net(lambda type, x: nln(type, x)) - # check_symbolic((soft, hard, symbolic), (training_data, test_data), trained_state) + num_experiments = 1 # 100 for paper + final_test_accuracies = [] + for i in range(num_experiments): + rng, int_rng, dropout_rng = jax.random.split(rng, 3) + trained_state, final_test_accuracy = train_and_evaluate( + int_rng, + dropout_rng, + soft, + (x_training, y_training, x_test, y_test), + get_config(), + ) + final_test_accuracies.append(final_test_accuracy) + print(f"{i}: final test accuracy: {final_test_accuracy * 100:.2f}") + # print mean, standard error of the mean, min, max, lowest 5%, highest 5% of final test accuracies + print( + f"mean: {numpy.mean(final_test_accuracies) * 100:.2f}, " + f"sem: {scipy.stats.sem(final_test_accuracies) * 100:.2f}, " + f"min: {numpy.min(final_test_accuracies) * 100:.2f}, " + f"max: {numpy.max(final_test_accuracies) * 100:.2f}, " + f"5%: {numpy.percentile(final_test_accuracies, 5) * 100:.2f}, " + f"95%: {numpy.percentile(final_test_accuracies, 95) * 100:.2f}" + ) + # numpy.set_printoptions(threshold=sys.maxsize) + # print(f"trained soft weights: {repr(trained_state.params)}") + # hard_weights = harden.hard_weights(trained_state.params) + # print(f"trained hard weights: {repr(hard_weights)}") + + """ + # Check symbolic net + _, hard, symbolic = neural_logic_net.net( + lambda type, x, training: nln(type, x, training) + ) + check_symbolic( + (soft, hard, symbolic), + (x_training, y_training, x_test, y_test), + trained_state, + dropout_rng, + ) + """ From 15bde550e8a6c2ffa2a49a0cceb661f7876ed4b0 Mon Sep 17 00:00:00 2001 From: Ian Wright <55286208+Z80coder@users.noreply.github.com> Date: Tue, 7 Feb 2023 13:01:56 +0000 Subject: [PATCH 008/113] *** 98% on noisy xor test set *** --- neurallogic/hard_majority.py | 15 ++- tests/test_noisy_xor.py | 239 ++++++----------------------------- 2 files changed, 55 insertions(+), 199 deletions(-) diff --git a/neurallogic/hard_majority.py b/neurallogic/hard_majority.py index 6c61eb2..ca9b31f 100644 --- a/neurallogic/hard_majority.py +++ b/neurallogic/hard_majority.py @@ -32,6 +32,7 @@ class SoftMajorityLayer(nn.Module): layer_size: The number of neurons in the layer. weights_init: The initializer function for the weight matrix. """ + @nn.compact def __call__(self, x): return soft_majority_layer(x) @@ -49,12 +50,22 @@ def __init__(self): def __call__(self, x): jaxpr = symbolic_generation.make_symbolic_flax_jaxpr( - self.hard_majority_layer, x) + self.hard_majority_layer, x + ) return symbolic_generation.symbolic_expression(jaxpr, x) majority_layer = neural_logic_net.select( lambda: SoftMajorityLayer(), lambda: HardMajorityLayer(), - lambda: SymbolicMajorityLayer() + lambda: SymbolicMajorityLayer(), ) + +# TODO: construct a majority-k generalisation of the above +# where k is the number of high-soft bits required for a majority +# and where k is a soft-bit parameter. Requires constructing +# a piecewise-continuous function (as per notebook). + + +# TODO: construct a soft-count layer from sorting/majority approach +# output is 1 high-soft bit that indicates the number of high-soft bits in the input diff --git a/tests/test_noisy_xor.py b/tests/test_noisy_xor.py index 759ebff..0522f99 100644 --- a/tests/test_noisy_xor.py +++ b/tests/test_noisy_xor.py @@ -89,203 +89,47 @@ def get_data(): return training_data, test_data -# 100% test accuracy (at some point) -def nln_1(type, x, training: bool): - x = hard_and.and_layer(type)(20)(x) - x = hard_not.not_layer(type)(4)(x) - x = x.ravel() - ######################################################## - x = harden_layer.harden_layer(type)(x) - x = x.reshape((num_classes, int(x.shape[0] / num_classes))) - x = x.sum(-1) - return x - - -# 77.83% -def nln_2(type, x, training: bool): - dtype = jax.numpy.float64 - layer_size = 64 # 64 - x = hard_and.and_layer(type)(layer_size)(x) - x = hard_or.or_layer(type)(layer_size)(x) - x = hard_not.not_layer(type)(4)(x) - x = x.ravel() - x = jax.numpy.array([x]) - x = hard_majority.majority_layer(type)()(x) - z = 1 - x - x = jax.numpy.concatenate([x, z], axis=0) - ######################################################## - # No need to harden here, since we're using majority for 2 classes - x = x.reshape((num_classes, int(x.shape[0] / num_classes))) - x = x.sum(-1) - return x - - -# 81.24 -def nln_3(type, x, training: bool): - y = jax.vmap(lambda x: 1 - x)(x) - x = jax.numpy.concatenate([x, y], axis=0) - dtype = jax.numpy.float64 - layer_size = 32 # 64 - x = hard_and.and_layer(type)(layer_size)(x) - x = hard_or.or_layer(type)(layer_size)(x) - x = hard_not.not_layer(type)(4)(x) - x = x.ravel() - x = jax.numpy.array([x]) - x = hard_majority.majority_layer(type)()(x) - z = 1 - x - x = jax.numpy.concatenate([x, z], axis=0) - ######################################################## - # No need to harden here, since we're using majority for 2 classes - x = x.reshape((num_classes, int(x.shape[0] / num_classes))) - x = x.sum(-1) - return x - - -# 82.30, but lots more high 90s -def nln_4(type, x, training: bool): - y = jax.vmap(lambda x: 1 - x)(x) - x = jax.numpy.concatenate([x, y], axis=0) - dtype = jax.numpy.float64 - layer_size = 64 # 64 - x = hard_and.and_layer(type)(layer_size)(x) - x = hard_or.or_layer(type)(layer_size)(x) - x = hard_not.not_layer(type)(8)(x) - x = x.ravel() - x = x.reshape((64, 8)) - x = hard_majority.majority_layer(type)()(x) - x = x.reshape((8, 8)) - x = hard_majority.majority_layer(type)()(x) - x = x.reshape((1, 8)) - x = hard_majority.majority_layer(type)()(x) - z = jax.vmap(lambda x: 1 - x)(x) - x = jax.numpy.concatenate([x, z], axis=0) - ######################################################## - x = x.reshape((num_classes, int(x.shape[0] / num_classes))) - x = x.sum(-1) - return x - - -# 93.26 -def nln_5(type, x, training: bool): - y = jax.vmap(lambda x: 1 - x)(x) - x = jax.numpy.concatenate([x, y], axis=0) - dtype = jax.numpy.float64 - layer_size = 128 # 64 - x = hard_and.and_layer(type)(layer_size)(x) - x = x.reshape((16, 8)) - x = hard_majority.majority_layer(type)()(x) - x = hard_not.not_layer(type)(8)(x) - x = x.reshape((16, 8)) - x = hard_majority.majority_layer(type)()(x) - x = hard_not.not_layer(type)(2)(x) - x = x.reshape((8, 4)) - x = hard_majority.majority_layer(type)()(x) - x = x.reshape((8, 1)) - x = hard_majority.majority_layer(type)()(x) - z = jax.vmap(lambda x: 1 - x)(x) - x = jax.numpy.concatenate([x, z], axis=0) - ######################################################## - x = x.reshape((num_classes, int(x.shape[0] / num_classes))) - x = x.sum(-1) - return x - - -# 93.90 -def nln_6(type, x, training: bool): - y = jax.vmap(lambda x: 1 - x)(x) - x = jax.numpy.concatenate([x, y], axis=0) - dtype = jax.numpy.float64 - layer_size = 128 - x = hard_and.and_layer(type)(layer_size)(x) - x = hard_not.not_layer(type)(8)(x) - x = x.reshape((64, 16)) - x = hard_majority.majority_layer(type)()(x) - x = hard_not.not_layer(type)(4)(x) - x = x.reshape((16, 16)) - x = hard_majority.majority_layer(type)()(x) - x = hard_not.not_layer(type)(2)(x) - x = x.reshape((8, 4)) - x = hard_majority.majority_layer(type)()(x) - x = x.reshape((8, 1)) - x = hard_majority.majority_layer(type)()(x) - z = jax.vmap(lambda x: 1 - x)(x) - x = jax.numpy.concatenate([x, z], axis=0) - ######################################################## - x = x.reshape((num_classes, int(x.shape[0] / num_classes))) - x = x.sum(-1) - return x - - -# 94.18 with peak_value=0.05 not 0.01 -def nln_7(type, x, training: bool): - y = jax.vmap(lambda x: 1 - x)(x) - x = jax.numpy.concatenate([x, y], axis=0) - dtype = jax.numpy.float64 - layer_size = 256 - x = hard_and.and_layer(type)(layer_size)(x) - x = hard_not.not_layer(type)(4)(x) - x = x.reshape((64, 16)) - x = hard_majority.majority_layer(type)()(x) - x = hard_not.not_layer(type)(4)(x) - x = x.reshape((16, 16)) - x = hard_majority.majority_layer(type)()(x) - x = hard_not.not_layer(type)(2)(x) - x = x.reshape((8, 4)) - x = hard_majority.majority_layer(type)()(x) - x = x.reshape((8, 1)) - x = hard_majority.majority_layer(type)()(x) - z = jax.vmap(lambda x: 1 - x)(x) - x = jax.numpy.concatenate([x, z], axis=0) - ######################################################## - x = x.reshape((num_classes, int(x.shape[0] / num_classes))) - x = x.sum(-1) - return x - - -# mean: 95.00, sem: 0.76, min: 68.84, max: 100.00, 5%: 75.72, 95%: 100.00 -def nln_8(type, x, training: bool): - y = jax.vmap(lambda x: 1 - x)(x) - x = jax.numpy.concatenate([x, y], axis=0) - dtype = jax.numpy.float64 - layer_size = 128 - x = hard_and.and_layer(type)(layer_size)(x) - x = hard_not.not_layer(type)(4)(x) - x = x.reshape((32, 16)) - x = hard_majority.majority_layer(type)()(x) - x = hard_not.not_layer(type)(4)(x) - x = x.reshape((16, 8)) - x = hard_majority.majority_layer(type)()(x) - x = hard_not.not_layer(type)(2)(x) - x = x.reshape((8, 4)) - x = hard_majority.majority_layer(type)()(x) - x = x.reshape((8, 1)) - x = hard_majority.majority_layer(type)()(x) - z = jax.vmap(lambda x: 1 - x)(x) - x = jax.numpy.concatenate([x, z], axis=0) - ######################################################## - x = x.reshape((num_classes, int(x.shape[0] / num_classes))) - x = x.sum(-1) - return x - - -# mean: 95.87, sem: 0.64, min: 64.28, max: 100.00, 5%: 80.49, 95%: 100.00 +""" +| Technique/Accuracy | Mean | 5 %ile | 95 %ile | Min | Max | +| ------------------- | -------------- | ------- | ------- | ------ | ------ | +| Tsetlin | 99.3 +/- 0.3 | 95.9 | 100.0 | 91.6 | 100.0 | +| dB | 98.0 +/- 0.4 | 93.9 | 100.0 | 70.3 | 100.0 | +| Neural network | 95.4 +/- 0.5 | 90.1 | 98.6 | 88.2 | 99.9 | +| SVM | 58.0 +/- 0.3 | 56.4 | 59.2 | 55.4 | 66.5 | +| Naive Bayes | 49.8 +/- 0.2 | 48.3 | 51.0 | 41.3 | 52.7 | +| Logistic regression | 49.8 +/- 0.3 | 47.8 | 51.1 | 41.1 | 53.1 | + +Source: https://arxiv.org/pdf/1804.01508.pdf +""" + +""" + schedule = optax.warmup_cosine_decay_schedule( + init_value=0.01, + peak_value=0.1, + warmup_steps=100, + decay_steps=config.num_epochs - 100, + end_value=0.05, + ) + tx = optax.chain( + # optax.clip(1.0), + #optax.adamw(learning_rate=schedule), + optax.adam(learning_rate=schedule), + ) +""" +# mean: 98.04, sem: 0.42, min: 70.32, max: 100.00, 5%: 93.90, 95%: 100.00 def nln(type, x, training: bool): y = jax.vmap(lambda x: 1 - x)(x) x = jax.numpy.concatenate([x, y], axis=0) - dtype = jax.numpy.float64 - layer_size = 128 - x = hard_and.and_layer(type)(layer_size)(x) - x = hard_not.not_layer(type)(4)(x) - x = x.reshape((32, 16)) - x = hard_majority.majority_layer(type)()(x) - x = hard_not.not_layer(type)(4)(x) - x = x.reshape((16, 8)) + dtype = jax.numpy.float32 + layer_size = 48 + x = hard_and.and_layer(type)(layer_size, dtype=dtype)(x) + x = hard_not.not_layer(type)(8, dtype=dtype)(x) # 48 x 8 = 384 + x = x.reshape((32, 12)) # 32 x 12 = 384 x = hard_majority.majority_layer(type)()(x) - x = hard_not.not_layer(type)(2)(x) - x = x.reshape((8, 4)) + x = x.reshape((4, 8)) # 4 x 8 = 32 x = hard_majority.majority_layer(type)()(x) - x = hard_not.not_layer(type)(1)(x) - x = x.reshape((8, 1)) + # Final majority should have reasonable width + x = x.reshape((4, 1)) # 4 x 1 = 4 x = hard_majority.majority_layer(type)()(x) z = jax.vmap(lambda x: 1 - x)(x) x = jax.numpy.concatenate([x, z], axis=0) @@ -325,14 +169,15 @@ def create_train_state(net, rng, dropout_rng, config): """ schedule = optax.warmup_cosine_decay_schedule( init_value=0.01, - peak_value=0.01, + peak_value=0.1, warmup_steps=100, decay_steps=config.num_epochs - 100, - end_value=0.01, + end_value=0.05, ) tx = optax.chain( # optax.clip(1.0), - optax.adamw(learning_rate=schedule), + # optax.adamw(learning_rate=schedule), + optax.adam(learning_rate=schedule), ) return TrainState.create( @@ -472,7 +317,7 @@ def get_config(): config.learning_rate = 0.01 config.momentum = 0.9 config.batch_size = 5000 - config.num_epochs = 800 + config.num_epochs = 1000 return config @@ -514,8 +359,8 @@ def test_noisy_xor(): # hard_weights = harden.hard_weights(trained_state.params) # print(f"trained hard weights: {repr(hard_weights)}") - """ # Check symbolic net + """ _, hard, symbolic = neural_logic_net.net( lambda type, x, training: nln(type, x, training) ) From 9cdd5107e8e11f823c217106a66f4b9b221d8608 Mon Sep 17 00:00:00 2001 From: Ian Wright <55286208+Z80coder@users.noreply.github.com> Date: Tue, 7 Feb 2023 15:06:23 +0000 Subject: [PATCH 009/113] *** 91% without hard-error on noisy xor *** --- tests/test_noisy_xor.py | 75 ++++++++++++++++++++++++++++++----------- 1 file changed, 55 insertions(+), 20 deletions(-) diff --git a/tests/test_noisy_xor.py b/tests/test_noisy_xor.py index 0522f99..a736f67 100644 --- a/tests/test_noisy_xor.py +++ b/tests/test_noisy_xor.py @@ -89,19 +89,6 @@ def get_data(): return training_data, test_data -""" -| Technique/Accuracy | Mean | 5 %ile | 95 %ile | Min | Max | -| ------------------- | -------------- | ------- | ------- | ------ | ------ | -| Tsetlin | 99.3 +/- 0.3 | 95.9 | 100.0 | 91.6 | 100.0 | -| dB | 98.0 +/- 0.4 | 93.9 | 100.0 | 70.3 | 100.0 | -| Neural network | 95.4 +/- 0.5 | 90.1 | 98.6 | 88.2 | 99.9 | -| SVM | 58.0 +/- 0.3 | 56.4 | 59.2 | 55.4 | 66.5 | -| Naive Bayes | 49.8 +/- 0.2 | 48.3 | 51.0 | 41.3 | 52.7 | -| Logistic regression | 49.8 +/- 0.3 | 47.8 | 51.1 | 41.1 | 53.1 | - -Source: https://arxiv.org/pdf/1804.01508.pdf -""" - """ schedule = optax.warmup_cosine_decay_schedule( init_value=0.01, @@ -116,8 +103,12 @@ def get_data(): optax.adam(learning_rate=schedule), ) """ +# with (4,1) error: # mean: 98.04, sem: 0.42, min: 70.32, max: 100.00, 5%: 93.90, 95%: 100.00 -def nln(type, x, training: bool): + + +# mean: 82.63, sem: 1.41, min: 48.86, max: 100.00, 5%: 59.70, 95%: 100.00 +def nln_1(type, x, training: bool): y = jax.vmap(lambda x: 1 - x)(x) x = jax.numpy.concatenate([x, y], axis=0) dtype = jax.numpy.float32 @@ -129,7 +120,7 @@ def nln(type, x, training: bool): x = x.reshape((4, 8)) # 4 x 8 = 32 x = hard_majority.majority_layer(type)()(x) # Final majority should have reasonable width - x = x.reshape((4, 1)) # 4 x 1 = 4 + x = x.reshape((1, 4)) # 4 x 1 = 4 x = hard_majority.majority_layer(type)()(x) z = jax.vmap(lambda x: 1 - x)(x) x = jax.numpy.concatenate([x, z], axis=0) @@ -139,6 +130,49 @@ def nln(type, x, training: bool): return x +""" +| Technique/Accuracy | Mean | 5 %ile | 95 %ile | Min | Max | +| ------------------- | -------------- | ------- | ------- | ------ | ------ | +| Tsetlin | 99.3 +/- 0.3 | 95.9 | 100.0 | 91.6 | 100.0 | +| Neural network | 95.4 +/- 0.5 | 90.1 | 98.6 | 88.2 | 99.9 | +| dB | 91.1 +/- 1.2 | 48.9 | 98.3 | 48.9 | 99.2 | +| SVM | 58.0 +/- 0.3 | 56.4 | 59.2 | 55.4 | 66.5 | +| Naive Bayes | 49.8 +/- 0.2 | 48.3 | 51.0 | 41.3 | 52.7 | +| Logistic regression | 49.8 +/- 0.3 | 47.8 | 51.1 | 41.1 | 53.1 | + +Source: https://arxiv.org/pdf/1804.01508.pdf +""" + + +""" +SGD +config.learning_rate = 2.0 +config.momentum = 0.9 +config.batch_size = 5000 +config.num_epochs = 2000 +""" +# mean: 91.06, sem: 1.15, min: 48.86, max: 99.16, 5%: 48.86, 95%: 98.28 +def nln(type, x, training: bool): + y = jax.vmap(lambda x: 1 - x)(x) + x = jax.numpy.concatenate([x, y], axis=0) + dtype = jax.numpy.float32 + layer_size = 32 + x = hard_and.and_layer(type)(layer_size, dtype=dtype)(x) + x = hard_or.or_layer(type)(layer_size, dtype=dtype)(x) + x = hard_not.not_layer(type)(64, dtype=dtype)(x) # 32 x 64 = 2048 + x = x.reshape((64, 32)) # 64 x 32 = 2048 + x = hard_majority.majority_layer(type)()(x) + x = x.reshape((1, 64)) # 1 x 64 = 64 + x = hard_majority.majority_layer(type)()(x) + z = jax.vmap(lambda x: 1 - x)(x) + x = jax.numpy.concatenate([x, z], axis=0) + ######################################################## + # x = harden_layer.harden_layer(type)(x) + x = x.reshape((num_classes, int(x.shape[0] / num_classes))) + x = x.sum(-1) + return x + + def batch_nln(type, x, training: bool): return jax.vmap(lambda x: nln(type, x, training))(x) @@ -169,15 +203,16 @@ def create_train_state(net, rng, dropout_rng, config): """ schedule = optax.warmup_cosine_decay_schedule( init_value=0.01, - peak_value=0.1, + peak_value=0.01, warmup_steps=100, decay_steps=config.num_epochs - 100, - end_value=0.05, + end_value=0.01, ) tx = optax.chain( # optax.clip(1.0), # optax.adamw(learning_rate=schedule), - optax.adam(learning_rate=schedule), + # optax.adam(learning_rate=schedule), + optax.sgd(learning_rate=config.learning_rate, momentum=config.momentum), ) return TrainState.create( @@ -314,10 +349,10 @@ def apply_hard_model_to_data(state, features, labels): def get_config(): config = ml_collections.ConfigDict() - config.learning_rate = 0.01 + config.learning_rate = 2.0 config.momentum = 0.9 config.batch_size = 5000 - config.num_epochs = 1000 + config.num_epochs = 2000 return config From f347c9191bb2aae2ac2de2d872f5d737a49ecd88 Mon Sep 17 00:00:00 2001 From: Ian Wright <55286208+Z80coder@users.noreply.github.com> Date: Thu, 9 Feb 2023 12:16:55 +0000 Subject: [PATCH 010/113] experimental variant of hard majority --- neurallogic/hard_majority.py | 22 +++++++++++++++++++++- 1 file changed, 21 insertions(+), 1 deletion(-) diff --git a/neurallogic/hard_majority.py b/neurallogic/hard_majority.py index ca9b31f..79e7389 100644 --- a/neurallogic/hard_majority.py +++ b/neurallogic/hard_majority.py @@ -8,12 +8,32 @@ def majority_index(input_size: int) -> int: return (input_size - 1) // 2 -def soft_majority(x: jax.numpy.array) -> float: +def soft_majority_1(x: jax.numpy.array) -> float: index = majority_index(x.shape[-1]) sorted_x = jax.numpy.sort(x, axis=-1) return jax.numpy.take(sorted_x, index, axis=-1) +def soft_majority(x: jax.numpy.array) -> float: + index = majority_index(x.shape[-1]) + sorted_x = jax.numpy.sort(x, axis=-1) + majority_bit = jax.numpy.take(sorted_x, index, axis=-1) + # print(f"majority_bit: {majority_bit}") + margin = jax.numpy.abs(majority_bit - 0.5) + # print(f"margin: {margin}") + mean = jax.numpy.mean(x, axis=-1) + # print(f"mean: {mean}") + margin_delta = mean * margin + # print(f"margin_delta: {margin_delta}") + representative_bit = jax.numpy.where( + majority_bit > 0.5, + 0.5 + margin_delta, + majority_bit + margin_delta, + ) + # print(f"representative_bit: {representative_bit}") + return representative_bit + + def hard_majority(x: jax.numpy.array) -> bool: threshold = x.shape[-1] - majority_index(x.shape[-1]) return jax.numpy.sum(x, axis=-1) >= threshold From dd2437cba354e0f6a7d0de17f1acca9d98e6f82c Mon Sep 17 00:00:00 2001 From: Ian Wright <55286208+Z80coder@users.noreply.github.com> Date: Thu, 9 Feb 2023 12:17:34 +0000 Subject: [PATCH 011/113] experimental mask variants --- neurallogic/hard_masks.py | 15 ++++++++++++++- 1 file changed, 14 insertions(+), 1 deletion(-) diff --git a/neurallogic/hard_masks.py b/neurallogic/hard_masks.py index 702ac61..5fb3638 100644 --- a/neurallogic/hard_masks.py +++ b/neurallogic/hard_masks.py @@ -3,7 +3,7 @@ import jax from flax import linen as nn -from neurallogic import neural_logic_net, symbolic_generation +from neurallogic import neural_logic_net, symbolic_generation, hard_not def soft_mask_to_true(w: float, x: float) -> float: @@ -18,6 +18,14 @@ def soft_mask_to_true(w: float, x: float) -> float: return jax.numpy.maximum(x, 1.0 - w) +def soft_mask_to_true_alt(w: float, b: float) -> float: + return jax.numpy.where( + w > 0.5, + jax.numpy.where(b > 0.5, b, (2 * w - 1) * b + 1 - w), + jax.numpy.where(b > 0.5, -2 * w * (1 - b) + 1, 1 - w), + ) + + def hard_mask_to_true(w, x): return jax.numpy.logical_or(x, jax.numpy.logical_not(w)) @@ -44,6 +52,11 @@ def soft_mask_to_false(w: float, x: float) -> float: return 1.0 - jax.numpy.maximum(1.0 - x, 1.0 - w) +# 1 - DifferentiableHardAND[1-b, w] +def soft_mask_to_false_alt(w: float, b: float) -> float: + return 1 - soft_mask_to_true(1 - b, w) + + def hard_mask_to_false(w, x): return jax.numpy.logical_and(x, w) From 230f4c861f9f939740c8480b97d34976f8daf9ba Mon Sep 17 00:00:00 2001 From: Ian Wright <55286208+Z80coder@users.noreply.github.com> Date: Thu, 9 Feb 2023 12:17:55 +0000 Subject: [PATCH 012/113] reformat --- neurallogic/hard_not.py | 20 ++++++++++++++++---- 1 file changed, 16 insertions(+), 4 deletions(-) diff --git a/neurallogic/hard_not.py b/neurallogic/hard_not.py index 3c2da45..0b80d39 100644 --- a/neurallogic/hard_not.py +++ b/neurallogic/hard_not.py @@ -5,6 +5,10 @@ from neurallogic import neural_logic_net, symbolic_generation +# TODO: replace hard-clip in layers with this +def logistic_clip(x): + return jax.scipy.special.expit(3 * (2 * x - 1)) + def soft_not(w: float, x: float) -> float: """ @@ -40,7 +44,9 @@ class SoftNotLayer(nn.Module): @nn.compact def __call__(self, x): weights_shape = (self.layer_size, jax.numpy.shape(x)[-1]) - weights = self.param("bit_weights", self.weights_init, weights_shape, self.dtype) + weights = self.param( + "bit_weights", self.weights_init, weights_shape, self.dtype + ) x = jax.numpy.asarray(x, self.dtype) return soft_not_layer(weights, x) @@ -67,7 +73,13 @@ def __call__(self, x): not_layer = neural_logic_net.select( - lambda layer_size, weights_init=nn.initializers.uniform(1.0), dtype=jax.numpy.float32: SoftNotLayer(layer_size, weights_init, dtype), - lambda layer_size, weights_init=nn.initializers.uniform(1.0), dtype=jax.numpy.float32: HardNotLayer(layer_size), - lambda layer_size, weights_init=nn.initializers.uniform(1.0), dtype=jax.numpy.float32: SymbolicNotLayer(layer_size), + lambda layer_size, weights_init=nn.initializers.uniform( + 1.0 + ), dtype=jax.numpy.float32: SoftNotLayer(layer_size, weights_init, dtype), + lambda layer_size, weights_init=nn.initializers.uniform( + 1.0 + ), dtype=jax.numpy.float32: HardNotLayer(layer_size), + lambda layer_size, weights_init=nn.initializers.uniform( + 1.0 + ), dtype=jax.numpy.float32: SymbolicNotLayer(layer_size), ) From adf7415b2ad2ff378ecab119f2c1e82cd6fe0e35 Mon Sep 17 00:00:00 2001 From: Ian Wright <55286208+Z80coder@users.noreply.github.com> Date: Thu, 9 Feb 2023 12:18:21 +0000 Subject: [PATCH 013/113] **** noisy xor bookmark: 94.15 % *** --- tests/test_noisy_xor.py | 230 +++++++++++++++++++++++++++++++++++----- 1 file changed, 201 insertions(+), 29 deletions(-) diff --git a/tests/test_noisy_xor.py b/tests/test_noisy_xor.py index a736f67..cb4c8d5 100644 --- a/tests/test_noisy_xor.py +++ b/tests/test_noisy_xor.py @@ -25,7 +25,7 @@ ) from tests import utils -# config.update("jax_enable_x64", True) +config.update("jax_enable_x64", True) def check_symbolic(nets, data, trained_state, dropout_rng): @@ -130,20 +130,6 @@ def nln_1(type, x, training: bool): return x -""" -| Technique/Accuracy | Mean | 5 %ile | 95 %ile | Min | Max | -| ------------------- | -------------- | ------- | ------- | ------ | ------ | -| Tsetlin | 99.3 +/- 0.3 | 95.9 | 100.0 | 91.6 | 100.0 | -| Neural network | 95.4 +/- 0.5 | 90.1 | 98.6 | 88.2 | 99.9 | -| dB | 91.1 +/- 1.2 | 48.9 | 98.3 | 48.9 | 99.2 | -| SVM | 58.0 +/- 0.3 | 56.4 | 59.2 | 55.4 | 66.5 | -| Naive Bayes | 49.8 +/- 0.2 | 48.3 | 51.0 | 41.3 | 52.7 | -| Logistic regression | 49.8 +/- 0.3 | 47.8 | 51.1 | 41.1 | 53.1 | - -Source: https://arxiv.org/pdf/1804.01508.pdf -""" - - """ SGD config.learning_rate = 2.0 @@ -152,7 +138,7 @@ def nln_1(type, x, training: bool): config.num_epochs = 2000 """ # mean: 91.06, sem: 1.15, min: 48.86, max: 99.16, 5%: 48.86, 95%: 98.28 -def nln(type, x, training: bool): +def nln_2(type, x, training: bool): y = jax.vmap(lambda x: 1 - x)(x) x = jax.numpy.concatenate([x, y], axis=0) dtype = jax.numpy.float32 @@ -167,7 +153,175 @@ def nln(type, x, training: bool): z = jax.vmap(lambda x: 1 - x)(x) x = jax.numpy.concatenate([x, z], axis=0) ######################################################## - # x = harden_layer.harden_layer(type)(x) + x = x.reshape((num_classes, int(x.shape[0] / num_classes))) + x = x.sum(-1) + return x + + +""" +SGD +config.learning_rate = 2.0 +config.momentum = 0.9 +config.batch_size = 5000 +config.num_epochs = 2000 +""" + +# mean: 91.54, sem: 0.64, min: 48.86, max: 99.58, 5%: 83.09, 95%: 97.10 +def nln_3(type, x, training: bool): + y = jax.vmap(lambda x: 1 - x)(x) + x = jax.numpy.concatenate([x, y], axis=0) + + dtype = jax.numpy.float32 + layer_size = 64 + x = hard_and.and_layer(type)(layer_size, dtype=dtype)(x) + x = hard_or.or_layer(type)(layer_size, dtype=dtype)(x) + x = hard_not.not_layer(type)(16, dtype=dtype)(x) + + x = x.reshape((1, layer_size * 16)) + x = hard_majority.majority_layer(type)()(x) + + z = jax.vmap(lambda x: 1 - x)(x) + x = jax.numpy.concatenate([x, z], axis=0) + ######################################################## + x = x.reshape((num_classes, int(x.shape[0] / num_classes))) + x = x.sum(-1) + return x + + +""" +SGD +config.learning_rate = 1.0 +config.momentum = 0.9 +config.batch_size = 5000 +config.num_epochs = 2500 +""" + +# mean: 91.84, sem: 0.98, min: 48.86, max: 99.66, 5%: 70.30, 95%: 98.79 +def nln_4(type, x, training: bool): + y = jax.vmap(lambda x: 1 - x)(x) + x = jax.numpy.concatenate([x, y], axis=0) + + dtype = jax.numpy.float32 + layer_size = 32 + x = hard_and.and_layer(type)(layer_size, dtype=dtype)(x) + x = hard_or.or_layer(type)(layer_size, dtype=dtype)(x) + x = hard_not.not_layer(type)(16, dtype=dtype)(x) + + x = x.reshape((1, layer_size * 16)) + x = hard_majority.majority_layer(type)()(x) + + z = jax.vmap(lambda x: 1 - x)(x) + x = jax.numpy.concatenate([x, z], axis=0) + ######################################################## + x = x.reshape((num_classes, int(x.shape[0] / num_classes))) + x = x.sum(-1) + return x + + +# mean: 89.77, sem: 1.18, min: 48.80, max: 100.00, 5%: 67.40, 95%: 99.04 +def nln_5(type, x, training: bool): + y = jax.vmap(lambda x: 1 - x)(x) + x = jax.numpy.concatenate([x, y], axis=0) + + dtype = jax.numpy.float32 + layer_size = 32 + x = hard_and.and_layer(type)(layer_size, dtype=dtype)(x) + x = hard_or.or_layer(type)(layer_size, dtype=dtype)(x) + x = hard_not.not_layer(type)(8, dtype=dtype)(x) + + x = x.reshape((1, layer_size * 8)) + x = hard_majority.majority_layer(type)()(x) + + z = jax.vmap(lambda x: 1 - x)(x) + x = jax.numpy.concatenate([x, z], axis=0) + ######################################################## + x = x.reshape((num_classes, int(x.shape[0] / num_classes))) + x = x.sum(-1) + return x + + +# mean: 89.19, sem: 1.42, min: 48.86, max: 100.00, 5%: 51.14, 95%: 98.86 +def nln_6(type, x, training: bool): + y = jax.vmap(lambda x: 1 - x)(x) + x = jax.numpy.concatenate([x, y], axis=0) + + dtype = jax.numpy.float32 + layer_size = 32 + x = hard_and.and_layer(type)(layer_size, dtype=dtype)(x) + x = hard_or.or_layer(type)(layer_size, dtype=dtype)(x) + x = hard_not.not_layer(type)(16, dtype=dtype)(x) + + x = x.reshape((16, layer_size)) + x = hard_majority.majority_layer(type)()(x) + + x = hard_not.not_layer(type)(8, dtype=dtype)(x) + x = x.reshape((1, 16 * 8)) + x = hard_majority.majority_layer(type)()(x) + + z = jax.vmap(lambda x: 1 - x)(x) + x = jax.numpy.concatenate([x, z], axis=0) + ######################################################## + x = x.reshape((num_classes, int(x.shape[0] / num_classes))) + x = x.sum(-1) + return x + + +# mean: 89.34, sem: 0.74, min: 51.16, max: 99.28, 5%: 76.88, 95%: 96.98 +def nln_7(type, x, training: bool): + y = jax.vmap(lambda x: 1 - x)(x) + x = jax.numpy.concatenate([x, y], axis=0) + + dtype = jax.numpy.float64 + layer_size = 40 + x = hard_and.and_layer(type)(layer_size, dtype=dtype)(x) + x = hard_or.or_layer(type)(layer_size, dtype=dtype)(x) + x = hard_not.not_layer(type)(20, dtype=dtype)(x) + + x = x.reshape((1, layer_size * 20)) + x = hard_majority.majority_layer(type)()(x) + + z = jax.vmap(lambda x: 1 - x)(x) + x = jax.numpy.concatenate([x, z], axis=0) + ######################################################## + x = x.reshape((num_classes, int(x.shape[0] / num_classes))) + x = x.sum(-1) + return x + + +""" +| Technique/Accuracy | Mean | 5 %ile | 95 %ile | Min | Max | +| ------------------- | -------------- | ------- | ------- | ------ | ------ | +| Tsetlin | 99.3 +/- 0.3 | 95.9 | 100.0 | 91.6 | 100.0 | +| Neural network | 95.4 +/- 0.5 | 90.1 | 98.6 | 88.2 | 99.9 | +| dB | 94.2 +/- 0.44 | 85.1 | 99.1 | 74.6 | 100.0 | +| SVM | 58.0 +/- 0.3 | 56.4 | 59.2 | 55.4 | 66.5 | +| Naive Bayes | 49.8 +/- 0.2 | 48.3 | 51.0 | 41.3 | 52.7 | +| Logistic regression | 49.8 +/- 0.3 | 47.8 | 51.1 | 41.1 | 53.1 | + +Source: https://arxiv.org/pdf/1804.01508.pdf +""" + +# mean: 94.15, sem: 0.44, min: 74.62, max: 100.00, 5%: 85.08, 95%: 99.10 +# soft_majority variant +# optax.radam(learning_rate=0.005) +# config.batch_size = 5000 +# config.num_epochs = 3000 +def nln(type, x, training: bool): + y = jax.vmap(lambda x: 1 - x)(x) + x = jax.numpy.concatenate([x, y], axis=0) + + dtype = jax.numpy.float64 + layer_size = 32 + x = hard_and.and_layer(type)(layer_size, dtype=dtype)(x) + x = hard_or.or_layer(type)(layer_size, dtype=dtype)(x) + x = hard_not.not_layer(type)(16, dtype=dtype)(x) + + x = x.reshape((1, 16 * layer_size)) + x = hard_majority.majority_layer(type)()(x) + + z = jax.vmap(lambda x: 1 - x)(x) + x = jax.numpy.concatenate([x, z], axis=0) + ######################################################## x = x.reshape((num_classes, int(x.shape[0] / num_classes))) x = x.sum(-1) return x @@ -202,17 +356,26 @@ def create_train_state(net, rng, dropout_rng, config): ) """ schedule = optax.warmup_cosine_decay_schedule( - init_value=0.01, - peak_value=0.01, - warmup_steps=100, - decay_steps=config.num_epochs - 100, + init_value=0.1, + peak_value=2, + warmup_steps=0, + decay_steps=500, end_value=0.01, ) tx = optax.chain( # optax.clip(1.0), - # optax.adamw(learning_rate=schedule), - # optax.adam(learning_rate=schedule), - optax.sgd(learning_rate=config.learning_rate, momentum=config.momentum), + # optax.adamw(learning_rate=0.01, weight_decay=0.0001), + # optax.adam(learning_rate=config.learning_rate), + # optax.adabelief(learning_rate=0.01), + # optax.adamax(learning_rate=0.01), + # optax.amsgrad(learning_rate=0.002), + # optax.sm3(learning_rate=0.1), + # optax.novograd(learning_rate=0.001), + # optax.optimistic_gradient_descent(learning_rate=1), + optax.radam(learning_rate=0.005), # 0.02 + # optax.sgd( + # learning_rate=config.learning_rate, momentum=config.momentum, nesterov=False + # ), ) return TrainState.create( @@ -225,6 +388,15 @@ def update_model(state, grads): return state.apply_gradients(grads=grads) +def my_hinge_loss(predictor_outputs, targets): + loss = jax.numpy.abs(predictor_outputs - targets) + + def hinge(x): + return jax.numpy.where(x >= 0.4, x * x, 0) + + return jax.vmap(hinge)(loss) + + def apply_model_with_grad_impl(state, features, labels, dropout_rng, training: bool): dropout_train_rng = jax.random.fold_in(key=dropout_rng, data=state.step) @@ -235,9 +407,11 @@ def loss_fn(params): training=training, rngs={"dropout": dropout_train_rng}, ) - one_hot = jax.nn.one_hot(labels, num_classes) + one_hot = jax.nn.one_hot(labels, num_classes, dtype=jax.numpy.int32) loss = jax.numpy.mean( optax.softmax_cross_entropy(logits=logits, labels=one_hot) + # optax.l2_loss(predictions=logits, targets=one_hot) + # my_hinge_loss(logits, one_hot) ) return loss, logits @@ -349,10 +523,10 @@ def apply_hard_model_to_data(state, features, labels): def get_config(): config = ml_collections.ConfigDict() - config.learning_rate = 2.0 + config.learning_rate = 0.001 config.momentum = 0.9 config.batch_size = 5000 - config.num_epochs = 2000 + config.num_epochs = 3000 return config @@ -395,7 +569,6 @@ def test_noisy_xor(): # print(f"trained hard weights: {repr(hard_weights)}") # Check symbolic net - """ _, hard, symbolic = neural_logic_net.net( lambda type, x, training: nln(type, x, training) ) @@ -405,4 +578,3 @@ def test_noisy_xor(): trained_state, dropout_rng, ) - """ From 6f0873a32ad7055a138c882304fbc61a6fa9735b Mon Sep 17 00:00:00 2001 From: Ian Wright <55286208+Z80coder@users.noreply.github.com> Date: Thu, 9 Feb 2023 13:19:37 +0000 Subject: [PATCH 014/113] compare to standard majority, and deprecate it --- neurallogic/hard_majority.py | 2 +- tests/test_noisy_xor.py | 7 ++++++- 2 files changed, 7 insertions(+), 2 deletions(-) diff --git a/neurallogic/hard_majority.py b/neurallogic/hard_majority.py index 79e7389..e3c289d 100644 --- a/neurallogic/hard_majority.py +++ b/neurallogic/hard_majority.py @@ -8,7 +8,7 @@ def majority_index(input_size: int) -> int: return (input_size - 1) // 2 -def soft_majority_1(x: jax.numpy.array) -> float: +def soft_majority_deprecated(x: jax.numpy.array) -> float: index = majority_index(x.shape[-1]) sorted_x = jax.numpy.sort(x, axis=-1) return jax.numpy.take(sorted_x, index, axis=-1) diff --git a/tests/test_noisy_xor.py b/tests/test_noisy_xor.py index cb4c8d5..e43fc16 100644 --- a/tests/test_noisy_xor.py +++ b/tests/test_noisy_xor.py @@ -306,6 +306,9 @@ def nln_7(type, x, training: bool): # optax.radam(learning_rate=0.005) # config.batch_size = 5000 # config.num_epochs = 3000 +# N.B. With normal soft majority we get +# mean: 83.82, sem: 1.75, min: 48.86, max: 100.00, 5%: 48.86, 95%: 99.42 +# Hence new variant seems very effective in avoiding local minima def nln(type, x, training: bool): y = jax.vmap(lambda x: 1 - x)(x) x = jax.numpy.concatenate([x, y], axis=0) @@ -541,7 +544,7 @@ def test_noisy_xor(): rng = jax.random.PRNGKey(0) print(soft.tabulate(rng, x_training[0:1], training=False)) - num_experiments = 1 # 100 for paper + num_experiments = 100 # 100 for paper final_test_accuracies = [] for i in range(num_experiments): rng, int_rng, dropout_rng = jax.random.split(rng, 3) @@ -569,6 +572,7 @@ def test_noisy_xor(): # print(f"trained hard weights: {repr(hard_weights)}") # Check symbolic net + """ _, hard, symbolic = neural_logic_net.net( lambda type, x, training: nln(type, x, training) ) @@ -578,3 +582,4 @@ def test_noisy_xor(): trained_state, dropout_rng, ) + """ From c0ad8a01a659fb722031334efc022131d8128d63 Mon Sep 17 00:00:00 2001 From: Ian Wright <55286208+Z80coder@users.noreply.github.com> Date: Thu, 9 Feb 2023 13:20:42 +0000 Subject: [PATCH 015/113] only do 1 exp --- tests/test_noisy_xor.py | 4 +--- 1 file changed, 1 insertion(+), 3 deletions(-) diff --git a/tests/test_noisy_xor.py b/tests/test_noisy_xor.py index e43fc16..b0a3ae2 100644 --- a/tests/test_noisy_xor.py +++ b/tests/test_noisy_xor.py @@ -544,7 +544,7 @@ def test_noisy_xor(): rng = jax.random.PRNGKey(0) print(soft.tabulate(rng, x_training[0:1], training=False)) - num_experiments = 100 # 100 for paper + num_experiments = 1 # 100 for paper final_test_accuracies = [] for i in range(num_experiments): rng, int_rng, dropout_rng = jax.random.split(rng, 3) @@ -572,7 +572,6 @@ def test_noisy_xor(): # print(f"trained hard weights: {repr(hard_weights)}") # Check symbolic net - """ _, hard, symbolic = neural_logic_net.net( lambda type, x, training: nln(type, x, training) ) @@ -582,4 +581,3 @@ def test_noisy_xor(): trained_state, dropout_rng, ) - """ From abb2c8bc72b03cb69e053caff67154b64711ef2e Mon Sep 17 00:00:00 2001 From: Ian Wright <55286208+Z80coder@users.noreply.github.com> Date: Thu, 9 Feb 2023 14:22:42 +0000 Subject: [PATCH 016/113] update majority test --- neurallogic/hard_majority.py | 17 +- tests/test_hard_majority.py | 336 +++++++++++++++++++++++++---------- 2 files changed, 245 insertions(+), 108 deletions(-) diff --git a/neurallogic/hard_majority.py b/neurallogic/hard_majority.py index e3c289d..b0eec0f 100644 --- a/neurallogic/hard_majority.py +++ b/neurallogic/hard_majority.py @@ -8,29 +8,22 @@ def majority_index(input_size: int) -> int: return (input_size - 1) // 2 -def soft_majority_deprecated(x: jax.numpy.array) -> float: +def majority_bit(x: jax.numpy.array) -> float: index = majority_index(x.shape[-1]) sorted_x = jax.numpy.sort(x, axis=-1) return jax.numpy.take(sorted_x, index, axis=-1) def soft_majority(x: jax.numpy.array) -> float: - index = majority_index(x.shape[-1]) - sorted_x = jax.numpy.sort(x, axis=-1) - majority_bit = jax.numpy.take(sorted_x, index, axis=-1) - # print(f"majority_bit: {majority_bit}") - margin = jax.numpy.abs(majority_bit - 0.5) - # print(f"margin: {margin}") + m_bit = majority_bit(x) + margin = jax.numpy.abs(m_bit - 0.5) mean = jax.numpy.mean(x, axis=-1) - # print(f"mean: {mean}") margin_delta = mean * margin - # print(f"margin_delta: {margin_delta}") representative_bit = jax.numpy.where( - majority_bit > 0.5, + m_bit > 0.5, 0.5 + margin_delta, - majority_bit + margin_delta, + m_bit + margin_delta, ) - # print(f"representative_bit: {representative_bit}") return representative_bit diff --git a/tests/test_hard_majority.py b/tests/test_hard_majority.py index 77b9f7a..e0a8b3e 100644 --- a/tests/test_hard_majority.py +++ b/tests/test_hard_majority.py @@ -20,50 +20,91 @@ def test_majority_index(): assert hard_majority.majority_index(12) == 5 -def test_soft_majority(): - assert hard_majority.soft_majority(numpy.array([1.0])) == 1.0 - assert hard_majority.soft_majority(numpy.array([2.0, 1.0])) == 1.0 - assert hard_majority.soft_majority(numpy.array([1.0, 3.0, 2.0])) == 2.0 - assert hard_majority.soft_majority( - numpy.array([2.0, 1.0, 4.0, 3.0])) == 2.0 - assert hard_majority.soft_majority( - numpy.array([1.0, 2.0, 3.0, 4.0, 5.0])) == 3.0 - assert hard_majority.soft_majority( - numpy.array([6.0, 3.0, 2.0, 4.0, 5.0, 1.0])) == 3.0 - assert hard_majority.soft_majority(numpy.array( - [7.0, 6.0, 5.0, 4.0, 3.0, 2.0, 1.0])) == 4.0 - assert hard_majority.soft_majority(numpy.array( - [2.0, 1.0, 4.0, 3.0, 6.0, 5.0, 8.0, 7.0])) == 4.0 - assert hard_majority.soft_majority(numpy.array( - [1.0, 2.0, 3.0, 5.0, 4.0, 6.0, 7.0, 9.0, 8.0])) == 5.0 - assert hard_majority.soft_majority(numpy.array( - [1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0])) == 5.0 - assert hard_majority.soft_majority(numpy.array( - [11.0, 10.0, 9.0, 8.0, 7.0, 6.0, 5.0, 4.0, 3.0, 2.0, 1.0])) == 6.0 - assert hard_majority.soft_majority(numpy.array( - [1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0])) == 6.0 +def test_majority_bit(): + assert hard_majority.majority_bit(numpy.array([1.0])) == 1.0 + assert hard_majority.majority_bit(numpy.array([2.0, 1.0])) == 1.0 + assert hard_majority.majority_bit(numpy.array([1.0, 3.0, 2.0])) == 2.0 + assert hard_majority.majority_bit(numpy.array([2.0, 1.0, 4.0, 3.0])) == 2.0 + assert hard_majority.majority_bit(numpy.array([1.0, 2.0, 3.0, 4.0, 5.0])) == 3.0 + assert ( + hard_majority.majority_bit(numpy.array([6.0, 3.0, 2.0, 4.0, 5.0, 1.0])) == 3.0 + ) + assert ( + hard_majority.majority_bit(numpy.array([7.0, 6.0, 5.0, 4.0, 3.0, 2.0, 1.0])) + == 4.0 + ) + assert ( + hard_majority.majority_bit( + numpy.array([2.0, 1.0, 4.0, 3.0, 6.0, 5.0, 8.0, 7.0]) + ) + == 4.0 + ) + assert ( + hard_majority.majority_bit( + numpy.array([1.0, 2.0, 3.0, 5.0, 4.0, 6.0, 7.0, 9.0, 8.0]) + ) + == 5.0 + ) + assert ( + hard_majority.majority_bit( + numpy.array([1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0]) + ) + == 5.0 + ) + assert ( + hard_majority.majority_bit( + numpy.array([11.0, 10.0, 9.0, 8.0, 7.0, 6.0, 5.0, 4.0, 3.0, 2.0, 1.0]) + ) + == 6.0 + ) + assert ( + hard_majority.majority_bit( + numpy.array([1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0]) + ) + == 6.0 + ) def test_hard_majority(): assert hard_majority.hard_majority(numpy.array([True])) == True assert hard_majority.hard_majority(numpy.array([False])) == False assert hard_majority.hard_majority(numpy.array([True, False])) == False - assert hard_majority.hard_majority( - numpy.array([False, True, False])) == False - assert hard_majority.hard_majority( - numpy.array([True, False, True, False])) == False - assert hard_majority.hard_majority(numpy.array( - [False, True, False, True, False])) == False - assert hard_majority.hard_majority(numpy.array( - [True, True, True, False, True, False])) == True - assert hard_majority.hard_majority(numpy.array( - [True, False, False, True, True, True, False])) == True - assert hard_majority.hard_majority(numpy.array( - [False, True, False, True, False, True, False, True])) == False - assert hard_majority.hard_majority(numpy.array( - [True, True, True, True, True, False, True, True, True])) == True - assert hard_majority.hard_majority(numpy.array( - [True, False, False, False, False, False, True, True, True, True])) == False + assert hard_majority.hard_majority(numpy.array([False, True, False])) == False + assert hard_majority.hard_majority(numpy.array([True, False, True, False])) == False + assert ( + hard_majority.hard_majority(numpy.array([False, True, False, True, False])) + == False + ) + assert ( + hard_majority.hard_majority(numpy.array([True, True, True, False, True, False])) + == True + ) + assert ( + hard_majority.hard_majority( + numpy.array([True, False, False, True, True, True, False]) + ) + == True + ) + assert ( + hard_majority.hard_majority( + numpy.array([False, True, False, True, False, True, False, True]) + ) + == False + ) + assert ( + hard_majority.hard_majority( + numpy.array([True, True, True, True, True, False, True, True, True]) + ) + == True + ) + assert ( + hard_majority.hard_majority( + numpy.array( + [True, False, False, False, False, False, True, True, True, True] + ) + ) + == False + ) def test_soft_and_hard_majority_equivalence(): @@ -77,88 +118,191 @@ def test_soft_and_hard_majority_equivalence(): def test_soft_majority_layer(): - assert numpy.all(hard_majority.soft_majority_layer( - numpy.array([[2.0, 1.0], [1.0, 2.0]])) == numpy.array([1.0, 1.0])) - assert numpy.all(hard_majority.soft_majority_layer(numpy.array( - [[1.0, 2.0, 3.0], [3.0, 2.0, 1.0]])) == numpy.array([2.0, 2.0])) - assert numpy.all(hard_majority.soft_majority_layer(numpy.array( - [[1.0, 2.0, 3.0, 4.0], [4.0, 3.0, 2.0, 1.0]])) == numpy.array([2.0, 2.0])) - assert numpy.all(hard_majority.soft_majority_layer(numpy.array( - [[1.0, 2.0, 3.0, 4.0, 5.0], [5.0, 4.0, 3.0, 2.0, 1.0]])) == numpy.array([3.0, 3.0])) - assert numpy.all(hard_majority.soft_majority_layer(numpy.array( - [[1.0, 2.0, 3.0, 4.0, 5.0, 6.0], [6.0, 5.0, 4.0, 3.0, 2.0, 1.0]])) == numpy.array([3.0, 3.0])) - assert numpy.all(hard_majority.soft_majority_layer(numpy.array( - [[1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0], [7.0, 6.0, 5.0, 4.0, 3.0, 2.0, 1.0]])) == numpy.array([4.0, 4.0])) - assert numpy.all(hard_majority.soft_majority_layer(numpy.array( - [[1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0], [8.0, 7.0, 6.0, 5.0, 4.0, 3.0, 2.0, 1.0]])) == numpy.array([4.0, 4.0])) - assert numpy.all(hard_majority.soft_majority_layer(numpy.array( - [[1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0], [9.0, 8.0, 7.0, 6.0, 5.0, 4.0, 3.0, 2.0, 1.0]])) == numpy.array([5.0, 5.0])) - assert numpy.all(hard_majority.soft_majority_layer(numpy.array( - [[1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0], [10.0, 9.0, 8.0, 7.0, 6.0, 5.0, 4.0, 3.0, 2.0, 1.0]])) == numpy.array([5.0, 5.0])) - assert numpy.all(hard_majority.soft_majority_layer(numpy.array( - [[1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0], [11.0, 10.0, 9.0, 8.0, 7.0, 6.0, 5.0, 4.0, 3.0, 2.0, 1.0]])) == numpy.array([6.0, 6.0])) - assert numpy.all(hard_majority.soft_majority_layer(numpy.array( - [[1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0], [12.0, 11.0, 10.0, 9.0, 8.0, 7.0, 6.0, 5.0, 4.0, 3.0, 2.0, 1.0]])) == numpy.array([6.0, 6.0])) + assert numpy.all( + hard_majority.soft_majority_layer(numpy.array([[0.0, 1.0], [1.0, 0.0]])) + == numpy.array([0.25, 0.25]) + ) + assert numpy.all( + hard_majority.soft_majority_layer( + numpy.array([[0.0, 1.0, 1.0], [1.0, 0.0, 0.0]]) + ) + == numpy.array([0.8333334, 0.16666667]) + ) + assert numpy.all( + hard_majority.soft_majority_layer( + numpy.array([[1.0, 0.0, 1.0, 0.0], [1.0, 0.0, 1.0, 1.0]]) + ) + == numpy.array([0.25, 0.875]) + ) + assert numpy.all( + hard_majority.soft_majority_layer( + numpy.array([[0.0, 1.0, 1.0, 0.0, 0.0], [1.0, 1.0, 0.0, 1.0, 1.0]]) + ) + == numpy.array([0.2, 0.9]) + ) + assert numpy.all( + hard_majority.soft_majority_layer( + numpy.array( + [[0.0, 1.0, 0.0, 1.0, 0.0, 1.0], [1.0, 1.0, 1.0, 1.0, 1.0, 0.0]] + ) + ) + == numpy.array([0.25, 0.9166667]) + ) + assert numpy.all( + hard_majority.soft_majority_layer( + numpy.array( + [ + [1.0, 0.0, 0.0, 0.0, 0.0, 0.8, 0.4], + [1.0, 0.9, 0.8, 0.45, 0.48, 0.51, 0.52], + ] + ) + ) + == numpy.array([0.15714286, 0.51331425]) + ) def test_hard_majority_layer(): - assert numpy.all(hard_majority.hard_majority_layer(numpy.array( - [[True, False], [False, True]])) == numpy.array([False, False])) - assert numpy.all(hard_majority.hard_majority_layer(numpy.array( - [[True, False, True], [True, False, False]])) == numpy.array([True, False])) - assert numpy.all(hard_majority.hard_majority_layer(numpy.array( - [[True, False, True, False], [False, True, False, True]])) == numpy.array([False, False])) - assert numpy.all(hard_majority.hard_majority_layer(numpy.array( - [[True, False, True, False, True], [True, False, True, False, True]])) == numpy.array([True, True])) - assert numpy.all(hard_majority.hard_majority_layer(numpy.array([[True, False, True, False, True, False], [ - False, True, False, True, False, True]])) == numpy.array([False, False])) - assert numpy.all(hard_majority.hard_majority_layer(numpy.array([[True, False, True, False, True, False, True], [ - True, False, True, False, True, False, False]])) == numpy.array([True, False])) - - assert numpy.all(hard_majority.hard_majority_layer(numpy.array([[True, False], [ - False, True], [False, True]])) == numpy.array([False, False, False])) - assert numpy.all(hard_majority.hard_majority_layer(numpy.array([[True, False, True], [ - True, False, True], [True, False, True]])) == numpy.array([True, True, True])) - assert numpy.all(hard_majority.hard_majority_layer(numpy.array([[True, False, True, False], [ - False, True, False, True], [False, True, False, True]])) == numpy.array([False, False, False])) - assert numpy.all(hard_majority.hard_majority_layer(numpy.array([[True, False, True, False, True], [ - True, False, True, False, True], [True, False, True, False, True]])) == numpy.array([True, True, True])) - assert numpy.all(hard_majority.hard_majority_layer(numpy.array([[True, False, True, False, True, False], [ - False, True, False, True, False, True], [False, True, False, True, False, True]])) == numpy.array([False, False, False])) + assert numpy.all( + hard_majority.hard_majority_layer(numpy.array([[True, False], [False, True]])) + == numpy.array([False, False]) + ) + assert numpy.all( + hard_majority.hard_majority_layer( + numpy.array([[True, False, True], [True, False, False]]) + ) + == numpy.array([True, False]) + ) + assert numpy.all( + hard_majority.hard_majority_layer( + numpy.array([[True, False, True, False], [False, True, False, True]]) + ) + == numpy.array([False, False]) + ) + assert numpy.all( + hard_majority.hard_majority_layer( + numpy.array( + [[True, False, True, False, True], [True, False, True, False, True]] + ) + ) + == numpy.array([True, True]) + ) + assert numpy.all( + hard_majority.hard_majority_layer( + numpy.array( + [ + [True, False, True, False, True, False], + [False, True, False, True, False, True], + ] + ) + ) + == numpy.array([False, False]) + ) + assert numpy.all( + hard_majority.hard_majority_layer( + numpy.array( + [ + [True, False, True, False, True, False, True], + [True, False, True, False, True, False, False], + ] + ) + ) + == numpy.array([True, False]) + ) + + assert numpy.all( + hard_majority.hard_majority_layer( + numpy.array([[True, False], [False, True], [False, True]]) + ) + == numpy.array([False, False, False]) + ) + assert numpy.all( + hard_majority.hard_majority_layer( + numpy.array([[True, False, True], [True, False, True], [True, False, True]]) + ) + == numpy.array([True, True, True]) + ) + assert numpy.all( + hard_majority.hard_majority_layer( + numpy.array( + [ + [True, False, True, False], + [False, True, False, True], + [False, True, False, True], + ] + ) + ) + == numpy.array([False, False, False]) + ) + assert numpy.all( + hard_majority.hard_majority_layer( + numpy.array( + [ + [True, False, True, False, True], + [True, False, True, False, True], + [True, False, True, False, True], + ] + ) + ) + == numpy.array([True, True, True]) + ) + assert numpy.all( + hard_majority.hard_majority_layer( + numpy.array( + [ + [True, False, True, False, True, False], + [False, True, False, True, False, True], + [False, True, False, True, False, True], + ] + ) + ) + == numpy.array([False, False, False]) + ) def test_layer(): test_data = [ - [ - [[0.8, 0.1, 0.4], [1.0, 0.0, 0.3]], - [0.4, 0.3] - ], + [[[0.8, 0.1, 0.4], [1.0, 0.0, 0.3]], [0.44333333, 0.3866667]], [ [[0.8, 0.1, 0.4], [1.0, 0.0, 0.3], [0.0, 0.0, 0.0]], - [0.4, 0.3, 0.0] + [0.44333333, 0.3866667, 0.0], ], [ [[0.8, 0.1, 0.4], [1.0, 0.0, 0.3], [0.8, 0.9, 0.1], [0.2, 0.01, 0.45]], - [0.4, 0.3, 0.8, 0.2] + [0.44333333, 0.3866667, 0.68, 0.266], ], [ - [[0.8, 0.1, 0.4], [1.0, 0.0, 0.3], [0.8, 0.9, 0.1], [0.2, 0.01, 0.45], [0.0, 0.0, 0.0]], - [0.4, 0.3, 0.8, 0.2, 0.0] + [ + [0.8, 0.1, 0.4], + [1.0, 0.0, 0.3], + [0.8, 0.9, 0.1], + [0.2, 0.01, 0.45], + [0.0, 0.0, 0.0], + ], + [0.44333333, 0.3866667, 0.68, 0.266, 0.0], ], [ - [[0.3, 0.93, 0.01, 0.5], [0.2, 0.01, 0.45, 0.1], [0.8, 0.9, 0.1, 0.2], [0.8, 0.1, 0.4, 0.3], [0.0, 0.0, 0.0, 0.0]], - [0.3, 0.1, 0.2, 0.3, 0.0] - ] + [ + [0.3, 0.93, 0.01, 0.5], + [0.2, 0.01, 0.45, 0.1], + [0.8, 0.9, 0.1, 0.2], + [0.8, 0.1, 0.4, 0.3], + [0.0, 0.0, 0.0, 0.0], + ], + [0.38700002, 0.176, 0.35000002, 0.38, 0.0], + ], ] for input, expected in test_data: + def soft(input): return hard_majority.soft_majority_layer(input) def hard(input): return hard_majority.hard_majority_layer(input) - utils.check_consistency(soft, hard, jax.numpy.array(expected), jax.numpy.array(input)) + utils.check_consistency( + soft, hard, jax.numpy.array(expected), jax.numpy.array(input) + ) -# TODO: test training the hard majority layer \ No newline at end of file +# TODO: test training the hard majority layer From dd0b71ff0eadab87906e0bc58233b18a82f533cc Mon Sep 17 00:00:00 2001 From: Ian Wright <55286208+Z80coder@users.noreply.github.com> Date: Fri, 10 Feb 2023 08:56:20 +0000 Subject: [PATCH 017/113] fix test for other archs --- tests/test_hard_majority.py | 36 ++++++++++++++++++------------------ 1 file changed, 18 insertions(+), 18 deletions(-) diff --git a/tests/test_hard_majority.py b/tests/test_hard_majority.py index e0a8b3e..08330cb 100644 --- a/tests/test_hard_majority.py +++ b/tests/test_hard_majority.py @@ -118,37 +118,37 @@ def test_soft_and_hard_majority_equivalence(): def test_soft_majority_layer(): - assert numpy.all( - hard_majority.soft_majority_layer(numpy.array([[0.0, 1.0], [1.0, 0.0]])) - == numpy.array([0.25, 0.25]) + assert numpy.allclose( + hard_majority.soft_majority_layer(numpy.array([[0.0, 1.0], [1.0, 0.0]])), + numpy.array([0.25, 0.25]), ) - assert numpy.all( + assert numpy.allclose( hard_majority.soft_majority_layer( numpy.array([[0.0, 1.0, 1.0], [1.0, 0.0, 0.0]]) - ) - == numpy.array([0.8333334, 0.16666667]) + ), + numpy.array([0.8333334, 0.16666667]), ) - assert numpy.all( + assert numpy.allclose( hard_majority.soft_majority_layer( numpy.array([[1.0, 0.0, 1.0, 0.0], [1.0, 0.0, 1.0, 1.0]]) - ) - == numpy.array([0.25, 0.875]) + ), + numpy.array([0.25, 0.875]), ) - assert numpy.all( + assert numpy.allclose( hard_majority.soft_majority_layer( numpy.array([[0.0, 1.0, 1.0, 0.0, 0.0], [1.0, 1.0, 0.0, 1.0, 1.0]]) - ) - == numpy.array([0.2, 0.9]) + ), + numpy.array([0.2, 0.9]), ) - assert numpy.all( + assert numpy.allclose( hard_majority.soft_majority_layer( numpy.array( [[0.0, 1.0, 0.0, 1.0, 0.0, 1.0], [1.0, 1.0, 1.0, 1.0, 1.0, 0.0]] ) - ) - == numpy.array([0.25, 0.9166667]) + ), + numpy.array([0.25, 0.9166667]), ) - assert numpy.all( + assert numpy.allclose( hard_majority.soft_majority_layer( numpy.array( [ @@ -156,8 +156,8 @@ def test_soft_majority_layer(): [1.0, 0.9, 0.8, 0.45, 0.48, 0.51, 0.52], ] ) - ) - == numpy.array([0.15714286, 0.51331425]) + ), + numpy.array([0.15714286, 0.51331425]), ) From 94bdc0cfe7efebd9551469c730827640dd50d6fb Mon Sep 17 00:00:00 2001 From: Ian Wright <55286208+Z80coder@users.noreply.github.com> Date: Fri, 10 Feb 2023 15:31:42 +0000 Subject: [PATCH 018/113] improved hardening layer --- neurallogic/hard_not.py | 4 ---- neurallogic/hard_xor.py | 9 +++++---- neurallogic/harden_layer.py | 26 ++++++++++++++++++++++---- 3 files changed, 27 insertions(+), 12 deletions(-) diff --git a/neurallogic/hard_not.py b/neurallogic/hard_not.py index 0b80d39..495682d 100644 --- a/neurallogic/hard_not.py +++ b/neurallogic/hard_not.py @@ -5,10 +5,6 @@ from neurallogic import neural_logic_net, symbolic_generation -# TODO: replace hard-clip in layers with this -def logistic_clip(x): - return jax.scipy.special.expit(3 * (2 * x - 1)) - def soft_not(w: float, x: float) -> float: """ diff --git a/neurallogic/hard_xor.py b/neurallogic/hard_xor.py index 460d1b5..fb557e0 100644 --- a/neurallogic/hard_xor.py +++ b/neurallogic/hard_xor.py @@ -6,15 +6,16 @@ from neurallogic import neural_logic_net, symbolic_generation, hard_masks +def differentiable_xor(x, y): + return jax.numpy.minimum(jax.numpy.maximum(x, y), 1.0 - jax.numpy.minimum(x, y)) + + # TODO: seperate out the mask from the xor operation def soft_xor_neuron(w, x): # Conditionally include input bits, according to weights x = jax.vmap(hard_masks.soft_mask_to_false, 0, 0)(w, x) - def xor(x, y): - return jax.numpy.minimum(jax.numpy.maximum(x, y), 1.0 - jax.numpy.minimum(x, y)) - - x = jax.lax.reduce(x, jax.numpy.array(0, dtype=x.dtype), xor, (0,)) + x = jax.lax.reduce(x, jax.numpy.array(0, dtype=x.dtype), differentiable_xor, (0,)) return x diff --git a/neurallogic/harden_layer.py b/neurallogic/harden_layer.py index 0024a98..5c82c18 100644 --- a/neurallogic/harden_layer.py +++ b/neurallogic/harden_layer.py @@ -3,19 +3,36 @@ from neurallogic import neural_logic_net -def harden_element(x): +def logistic_clip(x): + return jax.scipy.special.expit(3 * (2 * x - 1)) + + +def harden(x): # non-differentiable return jax.lax.cond(x > 0.5, lambda _: 1.0, lambda _: 0.0, None) -def straight_through_harden_element(x): +def scaled_straight_through_harden(x): + # The harden operation is non-differentiable. Therefore we need to + # approximate with the straight-through estimator. + # Create an exactly-zero expression with Sterbenz lemma that has # an exactly-one gradient. zero = x - jax.lax.stop_gradient(x) - return zero + jax.lax.stop_gradient(harden_element(x)) + one = zero + jax.lax.stop_gradient(harden(x)) + + # However, the straight-through estimator discards information about the value of x. + # In consequence, backprogated errors are independent of whether x is close to + # the decision boundary at 0.5. This is undesirable because we only want to + # allocate soft weight resources to the region around the decision boundary. + # We therefore scale the gradient to be smaller at x=0.5 and unscaled at + # x=0.0 and x=1.0. In other words, we minimally update upstream weights in order + # to potentially flip the hard value of x. + scale_factor = (2 * (0.5 - x)) * (2 * (0.5 - x)) + 0.001 + return one * scale_factor -soft_harden_layer = jax.vmap(straight_through_harden_element) +soft_harden_layer = jax.vmap(scaled_straight_through_harden) def hard_harden_layer(x): @@ -31,3 +48,4 @@ def symbolic_harden_layer(x): harden_layer = neural_logic_net.select( soft_harden_layer, hard_harden_layer, symbolic_harden_layer ) + \ No newline at end of file From a971dd0273125e3aff8aa2b5965ba8be0de24a29 Mon Sep 17 00:00:00 2001 From: Ian Wright <55286208+Z80coder@users.noreply.github.com> Date: Tue, 21 Feb 2023 08:46:56 +0000 Subject: [PATCH 019/113] **** noisy xor bookmark: 94.5 % ** --- neurallogic/hard_and.py | 13 ++++++++++ neurallogic/hard_masks.py | 19 +++++++-------- neurallogic/hard_not.py | 28 +++++++++++++--------- neurallogic/hard_or.py | 12 ++++++++++ neurallogic/harden_layer.py | 17 ++++---------- tests/test_noisy_xor.py | 47 +++++++++++++++++++++++++++++++++---- tests/utils.py | 8 +++---- 7 files changed, 101 insertions(+), 43 deletions(-) diff --git a/neurallogic/hard_and.py b/neurallogic/hard_and.py index 2fd58b1..e73dcf2 100644 --- a/neurallogic/hard_and.py +++ b/neurallogic/hard_and.py @@ -12,6 +12,19 @@ def soft_and_neuron(w, x): return jax.numpy.min(x) +def soft_and(x, y): + m = jax.numpy.minimum(x, y) + return jax.numpy.where( + 2 * m > 1, + 0.5 + 0.5 * (x + y) * (m - 0.5), + m + 0.5 * (x + y) * (0.5 - m), + ) + + +def soft_and_neuron_deprecated(w, x): + x = jax.vmap(hard_masks.soft_mask_to_true, 0, 0)(w, x) + return jax.lax.reduce(x, 1.0, soft_and, [0]) + def hard_and_neuron(w, x): x = jax.vmap(hard_masks.hard_mask_to_true, 0, 0)(w, x) return jax.lax.reduce(x, True, jax.lax.bitwise_and, [0]) diff --git a/neurallogic/hard_masks.py b/neurallogic/hard_masks.py index 5fb3638..cd3206a 100644 --- a/neurallogic/hard_masks.py +++ b/neurallogic/hard_masks.py @@ -3,7 +3,7 @@ import jax from flax import linen as nn -from neurallogic import neural_logic_net, symbolic_generation, hard_not +from neurallogic import neural_logic_net, symbolic_generation, hard_and, hard_or def soft_mask_to_true(w: float, x: float) -> float: @@ -17,13 +17,10 @@ def soft_mask_to_true(w: float, x: float) -> float: w = jax.numpy.clip(w, 0.0, 1.0) return jax.numpy.maximum(x, 1.0 - w) +def soft_mask_to_true_deprecated(w: float, x: float) -> float: + w = jax.numpy.clip(w, 0.0, 1.0) + return hard_or.soft_or(x, 1.0 - w) -def soft_mask_to_true_alt(w: float, b: float) -> float: - return jax.numpy.where( - w > 0.5, - jax.numpy.where(b > 0.5, b, (2 * w - 1) * b + 1 - w), - jax.numpy.where(b > 0.5, -2 * w * (1 - b) + 1, 1 - w), - ) def hard_mask_to_true(w, x): @@ -49,12 +46,12 @@ def soft_mask_to_false(w: float, x: float) -> float: Corresponding hard logic: b AND w """ w = jax.numpy.clip(w, 0.0, 1.0) + # TODO: what is this madness? return 1.0 - jax.numpy.maximum(1.0 - x, 1.0 - w) - -# 1 - DifferentiableHardAND[1-b, w] -def soft_mask_to_false_alt(w: float, b: float) -> float: - return 1 - soft_mask_to_true(1 - b, w) +def soft_mask_to_false_deprecated(w: float, x: float) -> float: + w = jax.numpy.clip(w, 0.0, 1.0) + return hard_and.soft_and(x, w) def hard_mask_to_false(w, x): diff --git a/neurallogic/hard_not.py b/neurallogic/hard_not.py index 495682d..c42d02a 100644 --- a/neurallogic/hard_not.py +++ b/neurallogic/hard_not.py @@ -3,7 +3,7 @@ import jax from flax import linen as nn -from neurallogic import neural_logic_net, symbolic_generation +from neurallogic import neural_logic_net, symbolic_generation, hard_and, hard_or def soft_not(w: float, x: float) -> float: @@ -17,6 +17,11 @@ def soft_not(w: float, x: float) -> float: w = jax.numpy.clip(w, 0.0, 1.0) return 1.0 - w + x * (2.0 * w - 1.0) +def soft_not_deprecated(w: float, x: float) -> float: + w = jax.numpy.clip(w, 0.0, 1.0) + # (w && x) || (! w && ! x) + return hard_or.soft_or(hard_and.soft_and(w, x), hard_and.soft_and(1.0 - w, 1.0 - x)) + def hard_not(w: bool, x: bool) -> bool: return jax.numpy.logical_not(jax.numpy.logical_xor(x, w)) @@ -31,10 +36,17 @@ def hard_not(w: bool, x: bool) -> bool: hard_not_layer = jax.vmap(hard_not_neuron, (0, None), 0) +def initialize_uniform_range(lower=0, upper=1): + def init(key, shape, dtype): + dtype = jax.dtypes.canonicalize_dtype(dtype) + x = jax.random.uniform(key, shape, dtype, lower, upper) + return x + + return init class SoftNotLayer(nn.Module): layer_size: int - weights_init: Callable = nn.initializers.uniform(1.0) + weights_init: Callable = initialize_uniform_range(0.49, 0.51) dtype: jax.numpy.dtype = jax.numpy.float32 @nn.compact @@ -69,13 +81,7 @@ def __call__(self, x): not_layer = neural_logic_net.select( - lambda layer_size, weights_init=nn.initializers.uniform( - 1.0 - ), dtype=jax.numpy.float32: SoftNotLayer(layer_size, weights_init, dtype), - lambda layer_size, weights_init=nn.initializers.uniform( - 1.0 - ), dtype=jax.numpy.float32: HardNotLayer(layer_size), - lambda layer_size, weights_init=nn.initializers.uniform( - 1.0 - ), dtype=jax.numpy.float32: SymbolicNotLayer(layer_size), + lambda layer_size, weights_init=initialize_uniform_range(0.49, 0.51), dtype=jax.numpy.float32: SoftNotLayer(layer_size, weights_init, dtype), + lambda layer_size, weights_init=initialize_uniform_range(0.49, 0.51), dtype=jax.numpy.float32: HardNotLayer(layer_size), + lambda layer_size, weights_init=initialize_uniform_range(0.49, 0.51), dtype=jax.numpy.float32: SymbolicNotLayer(layer_size), ) diff --git a/neurallogic/hard_or.py b/neurallogic/hard_or.py index c99b490..fa4917d 100644 --- a/neurallogic/hard_or.py +++ b/neurallogic/hard_or.py @@ -11,6 +11,18 @@ def soft_or_neuron(w, x): x = jax.vmap(hard_masks.soft_mask_to_false, 0, 0)(w, x) return jax.numpy.max(x) +def soft_or(x, y): + m = jax.numpy.maximum(x, y) + return jax.numpy.where( + 2 * m > 1, + 0.5 + 0.5 * (x + y) * (m - 0.5), + m + 0.5 * (x + y) * (0.5 - m), + ) + + +def soft_or_neuron_deprecated(w, x): + x = jax.vmap(hard_masks.soft_mask_to_true, 0, 0)(w, x) + return jax.lax.reduce(x, 0.0, soft_or, [0]) def hard_or_neuron(w, x): x = jax.vmap(hard_masks.hard_mask_to_false, 0, 0)(w, x) diff --git a/neurallogic/harden_layer.py b/neurallogic/harden_layer.py index 5c82c18..255d93a 100644 --- a/neurallogic/harden_layer.py +++ b/neurallogic/harden_layer.py @@ -12,27 +12,18 @@ def harden(x): return jax.lax.cond(x > 0.5, lambda _: 1.0, lambda _: 0.0, None) -def scaled_straight_through_harden(x): +def straight_through_harden(x): # The harden operation is non-differentiable. Therefore we need to # approximate with the straight-through estimator. # Create an exactly-zero expression with Sterbenz lemma that has # an exactly-one gradient. zero = x - jax.lax.stop_gradient(x) - one = zero + jax.lax.stop_gradient(harden(x)) + grad_of_one = zero + jax.lax.stop_gradient(harden(x)) + return grad_of_one - # However, the straight-through estimator discards information about the value of x. - # In consequence, backprogated errors are independent of whether x is close to - # the decision boundary at 0.5. This is undesirable because we only want to - # allocate soft weight resources to the region around the decision boundary. - # We therefore scale the gradient to be smaller at x=0.5 and unscaled at - # x=0.0 and x=1.0. In other words, we minimally update upstream weights in order - # to potentially flip the hard value of x. - scale_factor = (2 * (0.5 - x)) * (2 * (0.5 - x)) + 0.001 - return one * scale_factor - -soft_harden_layer = jax.vmap(scaled_straight_through_harden) +soft_harden_layer = jax.vmap(straight_through_harden) def hard_harden_layer(x): diff --git a/tests/test_noisy_xor.py b/tests/test_noisy_xor.py index b0a3ae2..beb9287 100644 --- a/tests/test_noisy_xor.py +++ b/tests/test_noisy_xor.py @@ -309,7 +309,7 @@ def nln_7(type, x, training: bool): # N.B. With normal soft majority we get # mean: 83.82, sem: 1.75, min: 48.86, max: 100.00, 5%: 48.86, 95%: 99.42 # Hence new variant seems very effective in avoiding local minima -def nln(type, x, training: bool): +def nln_8(type, x, training: bool): y = jax.vmap(lambda x: 1 - x)(x) x = jax.numpy.concatenate([x, y], axis=0) @@ -329,6 +329,38 @@ def nln(type, x, training: bool): x = x.sum(-1) return x +""" +| Technique/Accuracy | Mean | 5 %ile | 95 %ile | Min | Max | +| ------------------- | -------------- | ------- | ------- | ------ | ------ | +| Tsetlin | 99.3 +/- 0.3 | 95.9 | 100.0 | 91.6 | 100.0 | +| Neural network | 95.4 +/- 0.5 | 90.1 | 98.6 | 88.2 | 99.9 | +| dB | 94.5 +/- 0.4 | 89.2 | 98.2 | 70.8 | 100.0 | +| SVM | 58.0 +/- 0.3 | 56.4 | 59.2 | 55.4 | 66.5 | +| Naive Bayes | 49.8 +/- 0.2 | 48.3 | 51.0 | 41.3 | 52.7 | +| Logistic regression | 49.8 +/- 0.3 | 47.8 | 51.1 | 41.1 | 53.1 | + +Source: https://arxiv.org/pdf/1804.01508.pdf +""" +# mean: 94.51, sem: 0.37, min: 70.82, max: 100.00, 5%: 89.24, 95%: 98.15 +def nln(type, x, training: bool): + y = jax.vmap(lambda x: 1 - x)(x) + x = jax.numpy.concatenate([x, y], axis=0) + + dtype = jax.numpy.float64 + layer_size = 32 + x = hard_and.and_layer(type)(layer_size, dtype=dtype)(x) + x = hard_or.or_layer(type)(layer_size, dtype=dtype)(x) + x = hard_not.not_layer(type)(16, dtype=dtype)(x) + + x = x.reshape((1, 16 * layer_size)) + x = hard_majority.majority_layer(type)()(x) + + z = jax.vmap(lambda x: 1 - x)(x) + x = jax.numpy.concatenate([x, z], axis=0) + ######################################################## + x = x.reshape((num_classes, int(x.shape[0] / num_classes))) + x = x.sum(-1) + return x def batch_nln(type, x, training: bool): return jax.vmap(lambda x: nln(type, x, training))(x) @@ -375,7 +407,7 @@ def create_train_state(net, rng, dropout_rng, config): # optax.sm3(learning_rate=0.1), # optax.novograd(learning_rate=0.001), # optax.optimistic_gradient_descent(learning_rate=1), - optax.radam(learning_rate=0.005), # 0.02 + optax.radam(learning_rate=0.01), # 0.02 # optax.sgd( # learning_rate=config.learning_rate, momentum=config.momentum, nesterov=False # ), @@ -395,7 +427,7 @@ def my_hinge_loss(predictor_outputs, targets): loss = jax.numpy.abs(predictor_outputs - targets) def hinge(x): - return jax.numpy.where(x >= 0.4, x * x, 0) + return jax.numpy.where(x < 0.5, x, x) return jax.vmap(hinge)(loss) @@ -529,7 +561,7 @@ def get_config(): config.learning_rate = 0.001 config.momentum = 0.9 config.batch_size = 5000 - config.num_epochs = 3000 + config.num_epochs = 4000 return config @@ -571,7 +603,13 @@ def test_noisy_xor(): # hard_weights = harden.hard_weights(trained_state.params) # print(f"trained hard weights: {repr(hard_weights)}") + #if final_test_accuracy < 0.85: + # print("Aborting due to poor performance") + # break + + # Check symbolic net + """ _, hard, symbolic = neural_logic_net.net( lambda type, x, training: nln(type, x, training) ) @@ -581,3 +619,4 @@ def test_noisy_xor(): trained_state, dropout_rng, ) + """ diff --git a/tests/utils.py b/tests/utils.py index 1f2a41e..21f9219 100644 --- a/tests/utils.py +++ b/tests/utils.py @@ -72,21 +72,21 @@ def make_symbolic(*args): def check_consistency(soft: Callable, hard: Callable, expected, *args): - # print(f'\nchecking consistency for {soft.__name__}') + print(f'\nchecking consistency for {soft.__name__}') # Check that the soft function performs as expected soft_output = soft(*args) - # print(f'Expected: {expected}, Actual soft_output: {repr(soft_output)}') + print(f'Expected: {expected}, Actual soft_output: {repr(soft_output)}') assert numpy.allclose(soft_output, expected, equal_nan=True) # Check that the hard function performs as expected hard_args = tuple([harden.harden(arg) for arg in args]) hard_expected = harden.harden(expected) hard_output = hard(*hard_args) - # print(f'Expected: {hard_expected}, Actual hard_output: {repr(hard_output)}') + print(f'Expected: {hard_expected}, Actual hard_output: {repr(hard_output)}') assert numpy.allclose(hard_output, hard_expected, equal_nan=True) # Check that the jaxpr performs as expected symbolic_f = symbolic_generation.make_symbolic_jaxpr(hard, *hard_args) symbolic_output = symbolic_generation.eval_symbolic(symbolic_f, *hard_args) - # print(f'Expected: {hard_expected}, Actual symbolic_output: {repr(symbolic_output)}') + print(f'Expected: {hard_expected}, Actual symbolic_output: {repr(symbolic_output)}') assert numpy.allclose(symbolic_output, hard_expected, equal_nan=True) From c8428c41b64085f0a5ab21555b65cff84714755c Mon Sep 17 00:00:00 2001 From: Ian Wright <55286208+Z80coder@users.noreply.github.com> Date: Tue, 21 Feb 2023 09:38:27 +0000 Subject: [PATCH 020/113] *** noisy xor bookmark: 96.5 % *** --- neurallogic/hard_and.py | 21 ++---------- neurallogic/hard_masks.py | 6 ++-- neurallogic/hard_not.py | 21 ++++-------- neurallogic/hard_or.py | 28 ++++++---------- neurallogic/hard_xor.py | 1 - neurallogic/real_encoder.py | 4 +-- tests/test_noisy_xor.py | 66 ++++++++++++++++++++++++++----------- 7 files changed, 71 insertions(+), 76 deletions(-) diff --git a/neurallogic/hard_and.py b/neurallogic/hard_and.py index e73dcf2..099915e 100644 --- a/neurallogic/hard_and.py +++ b/neurallogic/hard_and.py @@ -3,7 +3,7 @@ import jax from flax import linen as nn -from neurallogic import hard_masks, neural_logic_net, symbolic_generation +from neurallogic import hard_masks, neural_logic_net, symbolic_generation, initialization # TODO: seperate and operation from mask operation @@ -35,21 +35,6 @@ def hard_and_neuron(w, x): hard_and_layer = jax.vmap(hard_and_neuron, (0, None), 0) -# TODO: move initialization to separate file -# TODO: simplify initialization to avoid the need to specify a guassian mean and std -def initialize_near_to_zero(mean=-1, std=0.5): - # TODO: investigate better initialization - def init(key, shape, dtype): - dtype = jax.dtypes.canonicalize_dtype(dtype) - # Sample from standard normal distribution (zero mean, unit variance) - x = jax.random.normal(key, shape, dtype) - # Transform to a normal distribution with mean -1 and standard deviation 0.5 - x = std * x + mean - x = jax.numpy.clip(x, 0.001, 0.999) - return x - - return init - class SoftAndLayer(nn.Module): """ @@ -61,7 +46,7 @@ class SoftAndLayer(nn.Module): """ layer_size: int - weights_init: Callable = initialize_near_to_zero() + weights_init: Callable = initialization.initialize_near_to_zero() dtype: jax.numpy.dtype = jax.numpy.float32 @nn.compact @@ -105,7 +90,7 @@ def __call__(self, x): and_layer = neural_logic_net.select( - lambda layer_size, weights_init=initialize_near_to_zero(), dtype=jax.numpy.float32: SoftAndLayer( + lambda layer_size, weights_init=initialization.initialize_near_to_zero(), dtype=jax.numpy.float32: SoftAndLayer( layer_size, weights_init, dtype ), lambda layer_size, weights_init=nn.initializers.constant( diff --git a/neurallogic/hard_masks.py b/neurallogic/hard_masks.py index cd3206a..371024a 100644 --- a/neurallogic/hard_masks.py +++ b/neurallogic/hard_masks.py @@ -3,10 +3,10 @@ import jax from flax import linen as nn -from neurallogic import neural_logic_net, symbolic_generation, hard_and, hard_or +from neurallogic import neural_logic_net, symbolic_generation, hard_and, hard_or, initialization -def soft_mask_to_true(w: float, x: float) -> float: +def soft_mask_to_true(w: float, x: float): """ w > 0.5 implies the mask operation is inactive, else active @@ -37,7 +37,7 @@ def hard_mask_to_true(w, x): hard_mask_to_true_layer = jax.vmap(hard_mask_to_true_neuron, (0, None), 0) -def soft_mask_to_false(w: float, x: float) -> float: +def soft_mask_to_false(w: float, x: float): """ w > 0.5 implies the mask is inactive, else active diff --git a/neurallogic/hard_not.py b/neurallogic/hard_not.py index c42d02a..90c3347 100644 --- a/neurallogic/hard_not.py +++ b/neurallogic/hard_not.py @@ -3,10 +3,10 @@ import jax from flax import linen as nn -from neurallogic import neural_logic_net, symbolic_generation, hard_and, hard_or +from neurallogic import neural_logic_net, symbolic_generation, hard_and, hard_or, initialization -def soft_not(w: float, x: float) -> float: +def soft_not(w, x): """ w > 0.5 implies the not operation is inactive, else active @@ -23,7 +23,7 @@ def soft_not_deprecated(w: float, x: float) -> float: return hard_or.soft_or(hard_and.soft_and(w, x), hard_and.soft_and(1.0 - w, 1.0 - x)) -def hard_not(w: bool, x: bool) -> bool: +def hard_not(w: bool, x: bool): return jax.numpy.logical_not(jax.numpy.logical_xor(x, w)) @@ -36,17 +36,10 @@ def hard_not(w: bool, x: bool) -> bool: hard_not_layer = jax.vmap(hard_not_neuron, (0, None), 0) -def initialize_uniform_range(lower=0, upper=1): - def init(key, shape, dtype): - dtype = jax.dtypes.canonicalize_dtype(dtype) - x = jax.random.uniform(key, shape, dtype, lower, upper) - return x - - return init class SoftNotLayer(nn.Module): layer_size: int - weights_init: Callable = initialize_uniform_range(0.49, 0.51) + weights_init: Callable = initialization.initialize_uniform_range(0.49, 0.51) dtype: jax.numpy.dtype = jax.numpy.float32 @nn.compact @@ -81,7 +74,7 @@ def __call__(self, x): not_layer = neural_logic_net.select( - lambda layer_size, weights_init=initialize_uniform_range(0.49, 0.51), dtype=jax.numpy.float32: SoftNotLayer(layer_size, weights_init, dtype), - lambda layer_size, weights_init=initialize_uniform_range(0.49, 0.51), dtype=jax.numpy.float32: HardNotLayer(layer_size), - lambda layer_size, weights_init=initialize_uniform_range(0.49, 0.51), dtype=jax.numpy.float32: SymbolicNotLayer(layer_size), + lambda layer_size, weights_init=initialization.initialize_uniform_range(0.49, 0.51), dtype=jax.numpy.float32: SoftNotLayer(layer_size, weights_init, dtype), + lambda layer_size, weights_init=initialization.initialize_uniform_range(0.49, 0.51), dtype=jax.numpy.float32: HardNotLayer(layer_size), + lambda layer_size, weights_init=initialization.initialize_uniform_range(0.49, 0.51), dtype=jax.numpy.float32: SymbolicNotLayer(layer_size), ) diff --git a/neurallogic/hard_or.py b/neurallogic/hard_or.py index fa4917d..b5f5240 100644 --- a/neurallogic/hard_or.py +++ b/neurallogic/hard_or.py @@ -3,7 +3,12 @@ import jax from flax import linen as nn -from neurallogic import neural_logic_net, symbolic_generation, hard_masks +from neurallogic import ( + neural_logic_net, + symbolic_generation, + hard_masks, + initialization, +) # TODO: seperate out the or operation from the mask operation @@ -11,6 +16,7 @@ def soft_or_neuron(w, x): x = jax.vmap(hard_masks.soft_mask_to_false, 0, 0)(w, x) return jax.numpy.max(x) + def soft_or(x, y): m = jax.numpy.maximum(x, y) return jax.numpy.where( @@ -24,6 +30,7 @@ def soft_or_neuron_deprecated(w, x): x = jax.vmap(hard_masks.soft_mask_to_true, 0, 0)(w, x) return jax.lax.reduce(x, 0.0, soft_or, [0]) + def hard_or_neuron(w, x): x = jax.vmap(hard_masks.hard_mask_to_false, 0, 0)(w, x) return jax.lax.reduce(x, False, jax.lax.bitwise_or, [0]) @@ -33,25 +40,10 @@ def hard_or_neuron(w, x): hard_or_layer = jax.vmap(hard_or_neuron, (0, None), 0) -# TODO: investigate better initialization - - -def initialize_near_to_one(): - def init(key, shape, dtype): - dtype = jax.dtypes.canonicalize_dtype(dtype) - # Sample from standard normal distribution (zero mean, unit variance) - x = jax.random.normal(key, shape, dtype) - # Transform to a normal distribution with mean 1 and standard deviation 0.5 - x = 0.5 * x + 1 - x = jax.numpy.clip(x, 0.001, 0.999) - return x - - return init - class SoftOrLayer(nn.Module): layer_size: int - weights_init: Callable = initialize_near_to_one() + weights_init: Callable = initialization.initialize_near_to_one() dtype: jax.numpy.dtype = jax.numpy.float32 @nn.compact @@ -87,7 +79,7 @@ def __call__(self, x): or_layer = neural_logic_net.select( - lambda layer_size, weights_init=initialize_near_to_one(), dtype=jax.numpy.float32: SoftOrLayer( + lambda layer_size, weights_init=initialization.initialize_near_to_one(), dtype=jax.numpy.float32: SoftOrLayer( layer_size, weights_init, dtype ), lambda layer_size, weights_init=nn.initializers.constant( diff --git a/neurallogic/hard_xor.py b/neurallogic/hard_xor.py index fb557e0..b35e8ef 100644 --- a/neurallogic/hard_xor.py +++ b/neurallogic/hard_xor.py @@ -34,7 +34,6 @@ class SoftXorLayer(nn.Module): layer_size: int weights_init: Callable = ( nn.initializers.uniform(1.0) - # hard_and.initialize_near_to_zero() ) dtype: jax.numpy.dtype = jax.numpy.float32 diff --git a/neurallogic/real_encoder.py b/neurallogic/real_encoder.py index 32024b4..1613f4c 100644 --- a/neurallogic/real_encoder.py +++ b/neurallogic/real_encoder.py @@ -8,7 +8,7 @@ # TODO: implement a soft_real_decoder that can perhaps replace the port count approach -def soft_real_encoder(t: float, x: float) -> float: +def soft_real_encoder(t: float, x: float): eps = 0.0000001 # x should be in [0, 1] t = jax.numpy.clip(t, 0, 1) @@ -25,7 +25,7 @@ def soft_real_encoder(t: float, x: float) -> float: ) -def hard_real_encoder(t: float, x: float) -> bool: +def hard_real_encoder(t, x): # t and x must be floats return jax.numpy.where(soft_real_encoder(t, x) > 0.5, True, False) diff --git a/tests/test_noisy_xor.py b/tests/test_noisy_xor.py index beb9287..87e23a2 100644 --- a/tests/test_noisy_xor.py +++ b/tests/test_noisy_xor.py @@ -22,6 +22,7 @@ harden, harden_layer, neural_logic_net, + initialization, ) from tests import utils @@ -137,6 +138,8 @@ def nln_1(type, x, training: bool): config.batch_size = 5000 config.num_epochs = 2000 """ + + # mean: 91.06, sem: 1.15, min: 48.86, max: 99.16, 5%: 48.86, 95%: 98.28 def nln_2(type, x, training: bool): y = jax.vmap(lambda x: 1 - x)(x) @@ -166,6 +169,7 @@ def nln_2(type, x, training: bool): config.num_epochs = 2000 """ + # mean: 91.54, sem: 0.64, min: 48.86, max: 99.58, 5%: 83.09, 95%: 97.10 def nln_3(type, x, training: bool): y = jax.vmap(lambda x: 1 - x)(x) @@ -196,6 +200,7 @@ def nln_3(type, x, training: bool): config.num_epochs = 2500 """ + # mean: 91.84, sem: 0.98, min: 48.86, max: 99.66, 5%: 70.30, 95%: 98.79 def nln_4(type, x, training: bool): y = jax.vmap(lambda x: 1 - x)(x) @@ -288,18 +293,6 @@ def nln_7(type, x, training: bool): return x -""" -| Technique/Accuracy | Mean | 5 %ile | 95 %ile | Min | Max | -| ------------------- | -------------- | ------- | ------- | ------ | ------ | -| Tsetlin | 99.3 +/- 0.3 | 95.9 | 100.0 | 91.6 | 100.0 | -| Neural network | 95.4 +/- 0.5 | 90.1 | 98.6 | 88.2 | 99.9 | -| dB | 94.2 +/- 0.44 | 85.1 | 99.1 | 74.6 | 100.0 | -| SVM | 58.0 +/- 0.3 | 56.4 | 59.2 | 55.4 | 66.5 | -| Naive Bayes | 49.8 +/- 0.2 | 48.3 | 51.0 | 41.3 | 52.7 | -| Logistic regression | 49.8 +/- 0.3 | 47.8 | 51.1 | 41.1 | 53.1 | - -Source: https://arxiv.org/pdf/1804.01508.pdf -""" # mean: 94.15, sem: 0.44, min: 74.62, max: 100.00, 5%: 85.08, 95%: 99.10 # soft_majority variant @@ -329,27 +322,60 @@ def nln_8(type, x, training: bool): x = x.sum(-1) return x + + +# mean: 94.51, sem: 0.37, min: 70.82, max: 100.00, 5%: 89.24, 95%: 98.15 +def nln_9(type, x, training: bool): + y = jax.vmap(lambda x: 1 - x)(x) + x = jax.numpy.concatenate([x, y], axis=0) + + dtype = jax.numpy.float64 + layer_size = 32 + x = hard_and.and_layer(type)(layer_size, dtype=dtype)(x) + x = hard_or.or_layer(type)(layer_size, dtype=dtype)(x) + x = hard_not.not_layer(type)(16, dtype=dtype)(x) + + x = x.reshape((1, 16 * layer_size)) + x = hard_majority.majority_layer(type)()(x) + + z = jax.vmap(lambda x: 1 - x)(x) + x = jax.numpy.concatenate([x, z], axis=0) + ######################################################## + x = x.reshape((num_classes, int(x.shape[0] / num_classes))) + x = x.sum(-1) + return x + + """ | Technique/Accuracy | Mean | 5 %ile | 95 %ile | Min | Max | | ------------------- | -------------- | ------- | ------- | ------ | ------ | | Tsetlin | 99.3 +/- 0.3 | 95.9 | 100.0 | 91.6 | 100.0 | +| dB | 96.5 +/- 0.2 | 93.2 | 99.5 | 87.9 | 100.0 | | Neural network | 95.4 +/- 0.5 | 90.1 | 98.6 | 88.2 | 99.9 | -| dB | 94.5 +/- 0.4 | 89.2 | 98.2 | 70.8 | 100.0 | | SVM | 58.0 +/- 0.3 | 56.4 | 59.2 | 55.4 | 66.5 | | Naive Bayes | 49.8 +/- 0.2 | 48.3 | 51.0 | 41.3 | 52.7 | | Logistic regression | 49.8 +/- 0.3 | 47.8 | 51.1 | 41.1 | 53.1 | Source: https://arxiv.org/pdf/1804.01508.pdf """ -# mean: 94.51, sem: 0.37, min: 70.82, max: 100.00, 5%: 89.24, 95%: 98.15 + +# mean: 96.52, sem: 0.23, min: 87.86, max: 100.00, 5%: 93.16, 95%: 99.51 def nln(type, x, training: bool): y = jax.vmap(lambda x: 1 - x)(x) x = jax.numpy.concatenate([x, y], axis=0) dtype = jax.numpy.float64 layer_size = 32 - x = hard_and.and_layer(type)(layer_size, dtype=dtype)(x) - x = hard_or.or_layer(type)(layer_size, dtype=dtype)(x) + x = hard_and.and_layer(type)( + layer_size, + dtype=dtype, + weights_init=initialization.initialize_bernoulli(0.01, 0.1, 0.51), + )(x) + x = hard_or.or_layer(type)( + layer_size, + dtype=dtype, + weights_init=initialization.initialize_bernoulli(0.99, 0.49, 0.9), + )(x) x = hard_not.not_layer(type)(16, dtype=dtype)(x) x = x.reshape((1, 16 * layer_size)) @@ -362,6 +388,7 @@ def nln(type, x, training: bool): x = x.sum(-1) return x + def batch_nln(type, x, training: bool): return jax.vmap(lambda x: nln(type, x, training))(x) @@ -551,7 +578,7 @@ def logits_fn(params): def apply_hard_model_to_data(state, features, labels): accuracy = 0 - for (image, label) in tqdm(zip(features, labels), total=len(features)): + for image, label in tqdm(zip(features, labels), total=len(features)): accuracy += apply_hard_model(state, image, label) return accuracy / len(features) @@ -561,7 +588,7 @@ def get_config(): config.learning_rate = 0.001 config.momentum = 0.9 config.batch_size = 5000 - config.num_epochs = 4000 + config.num_epochs = 1000 return config @@ -603,11 +630,10 @@ def test_noisy_xor(): # hard_weights = harden.hard_weights(trained_state.params) # print(f"trained hard weights: {repr(hard_weights)}") - #if final_test_accuracy < 0.85: + # if final_test_accuracy < 0.85: # print("Aborting due to poor performance") # break - # Check symbolic net """ _, hard, symbolic = neural_logic_net.net( From 2c0a852b18b052a84a94fa7ba702578e77fdaf12 Mon Sep 17 00:00:00 2001 From: Ian Wright <55286208+Z80coder@users.noreply.github.com> Date: Tue, 21 Feb 2023 10:13:55 +0000 Subject: [PATCH 021/113] *** noisy xor bookmark: 97.1 % *** --- neurallogic/hard_masks.py | 8 ++++---- tests/test_noisy_xor.py | 39 +++++++++++++++++++++++++++++++++------ 2 files changed, 37 insertions(+), 10 deletions(-) diff --git a/neurallogic/hard_masks.py b/neurallogic/hard_masks.py index 371024a..4fed76a 100644 --- a/neurallogic/hard_masks.py +++ b/neurallogic/hard_masks.py @@ -6,7 +6,7 @@ from neurallogic import neural_logic_net, symbolic_generation, hard_and, hard_or, initialization -def soft_mask_to_true(w: float, x: float): +def soft_mask_to_true_deprecated(w: float, x: float): """ w > 0.5 implies the mask operation is inactive, else active @@ -17,7 +17,7 @@ def soft_mask_to_true(w: float, x: float): w = jax.numpy.clip(w, 0.0, 1.0) return jax.numpy.maximum(x, 1.0 - w) -def soft_mask_to_true_deprecated(w: float, x: float) -> float: +def soft_mask_to_true(w: float, x: float) -> float: w = jax.numpy.clip(w, 0.0, 1.0) return hard_or.soft_or(x, 1.0 - w) @@ -37,7 +37,7 @@ def hard_mask_to_true(w, x): hard_mask_to_true_layer = jax.vmap(hard_mask_to_true_neuron, (0, None), 0) -def soft_mask_to_false(w: float, x: float): +def soft_mask_to_false_deprecated(w: float, x: float): """ w > 0.5 implies the mask is inactive, else active @@ -49,7 +49,7 @@ def soft_mask_to_false(w: float, x: float): # TODO: what is this madness? return 1.0 - jax.numpy.maximum(1.0 - x, 1.0 - w) -def soft_mask_to_false_deprecated(w: float, x: float) -> float: +def soft_mask_to_false(w: float, x: float) -> float: w = jax.numpy.clip(w, 0.0, 1.0) return hard_and.soft_and(x, w) diff --git a/tests/test_noisy_xor.py b/tests/test_noisy_xor.py index 87e23a2..906e2af 100644 --- a/tests/test_noisy_xor.py +++ b/tests/test_noisy_xor.py @@ -293,7 +293,6 @@ def nln_7(type, x, training: bool): return x - # mean: 94.15, sem: 0.44, min: 74.62, max: 100.00, 5%: 85.08, 95%: 99.10 # soft_majority variant # optax.radam(learning_rate=0.005) @@ -323,7 +322,6 @@ def nln_8(type, x, training: bool): return x - # mean: 94.51, sem: 0.37, min: 70.82, max: 100.00, 5%: 89.24, 95%: 98.15 def nln_9(type, x, training: bool): y = jax.vmap(lambda x: 1 - x)(x) @@ -346,11 +344,41 @@ def nln_9(type, x, training: bool): return x + +# mean: 96.52, sem: 0.23, min: 87.86, max: 100.00, 5%: 93.16, 95%: 99.51 +def nln_10(type, x, training: bool): + y = jax.vmap(lambda x: 1 - x)(x) + x = jax.numpy.concatenate([x, y], axis=0) + + dtype = jax.numpy.float64 + layer_size = 32 + x = hard_and.and_layer(type)( + layer_size, + dtype=dtype, + weights_init=initialization.initialize_bernoulli(0.01, 0.1, 0.51), + )(x) + x = hard_or.or_layer(type)( + layer_size, + dtype=dtype, + weights_init=initialization.initialize_bernoulli(0.99, 0.49, 0.9), + )(x) + x = hard_not.not_layer(type)(16, dtype=dtype)(x) + + x = x.reshape((1, 16 * layer_size)) + x = hard_majority.majority_layer(type)()(x) + + z = jax.vmap(lambda x: 1 - x)(x) + x = jax.numpy.concatenate([x, z], axis=0) + ######################################################## + x = x.reshape((num_classes, int(x.shape[0] / num_classes))) + x = x.sum(-1) + return x + """ | Technique/Accuracy | Mean | 5 %ile | 95 %ile | Min | Max | | ------------------- | -------------- | ------- | ------- | ------ | ------ | | Tsetlin | 99.3 +/- 0.3 | 95.9 | 100.0 | 91.6 | 100.0 | -| dB | 96.5 +/- 0.2 | 93.2 | 99.5 | 87.9 | 100.0 | +| dB | 97.1 +/- 0.2 | 93.9 | 99.6 | 92.2 | 100.0 | | Neural network | 95.4 +/- 0.5 | 90.1 | 98.6 | 88.2 | 99.9 | | SVM | 58.0 +/- 0.3 | 56.4 | 59.2 | 55.4 | 66.5 | | Naive Bayes | 49.8 +/- 0.2 | 48.3 | 51.0 | 41.3 | 52.7 | @@ -358,8 +386,7 @@ def nln_9(type, x, training: bool): Source: https://arxiv.org/pdf/1804.01508.pdf """ - -# mean: 96.52, sem: 0.23, min: 87.86, max: 100.00, 5%: 93.16, 95%: 99.51 +# mean: 97.12, sem: 0.18, min: 92.18, max: 100.00, 5%: 93.90, 95%: 99.58 def nln(type, x, training: bool): y = jax.vmap(lambda x: 1 - x)(x) x = jax.numpy.concatenate([x, y], axis=0) @@ -588,7 +615,7 @@ def get_config(): config.learning_rate = 0.001 config.momentum = 0.9 config.batch_size = 5000 - config.num_epochs = 1000 + config.num_epochs = 2000 return config From e4186b3e73b38eea6afb5e765fc842e570922ca8 Mon Sep 17 00:00:00 2001 From: Ian Wright <55286208+Z80coder@users.noreply.github.com> Date: Tue, 21 Feb 2023 15:10:11 +0000 Subject: [PATCH 022/113] add this --- neurallogic/initialization.py | 48 +++++++++++++++++++++++++++++++++++ 1 file changed, 48 insertions(+) create mode 100644 neurallogic/initialization.py diff --git a/neurallogic/initialization.py b/neurallogic/initialization.py new file mode 100644 index 0000000..12ef3b2 --- /dev/null +++ b/neurallogic/initialization.py @@ -0,0 +1,48 @@ +import jax + + +def initialize_uniform_range(lower=0.0, upper=1.0): + def init(key, shape, dtype): + dtype = jax.dtypes.canonicalize_dtype(dtype) + x = jax.random.uniform(key, shape, dtype, lower, upper) + return x + + return init + + +def initialize_near_to_zero(mean=-1, std=0.5): + def init(key, shape, dtype): + dtype = jax.dtypes.canonicalize_dtype(dtype) + # Sample from standard normal distribution (zero mean, unit variance) + x = jax.random.normal(key, shape, dtype) + # Transform to a normal distribution with mean -1 and standard deviation 0.5 + x = std * x + mean + x = jax.numpy.clip(x, 0.001, 0.999) + return x + + return init + + +def initialize_near_to_one(): + def init(key, shape, dtype): + dtype = jax.dtypes.canonicalize_dtype(dtype) + # Sample from standard normal distribution (zero mean, unit variance) + x = jax.random.normal(key, shape, dtype) + # Transform to a normal distribution with mean 1 and standard deviation 0.5 + x = 0.5 * x + 1 + x = jax.numpy.clip(x, 0.001, 0.999) + return x + + return init + + +# TODO: get rid of symmetry +def initialize_bernoulli(p=0.5, low=0.001, high=0.999): + def init(key, shape, dtype): + x = jax.random.bernoulli(key, p, shape) + x = jax.random.bernoulli(key, p, shape) + x = jax.numpy.where(x, high, low) + x = jax.numpy.asarray(x, dtype) + return x + + return init From b0062a2a34a733be2d94d9a75bdd79678f302a3c Mon Sep 17 00:00:00 2001 From: Ian Wright <55286208+Z80coder@users.noreply.github.com> Date: Wed, 22 Feb 2023 12:57:25 +0000 Subject: [PATCH 023/113] *** noisy xor 97.9% bookmark *** --- neurallogic/hard_and.py | 2 +- neurallogic/hard_masks.py | 2 + neurallogic/hard_not.py | 1 + neurallogic/hard_or.py | 2 +- neurallogic/initialization.py | 12 +- tests/test_noisy_xor.py | 377 ++-------------------------------- 6 files changed, 33 insertions(+), 363 deletions(-) diff --git a/neurallogic/hard_and.py b/neurallogic/hard_and.py index 099915e..794930e 100644 --- a/neurallogic/hard_and.py +++ b/neurallogic/hard_and.py @@ -20,7 +20,7 @@ def soft_and(x, y): m + 0.5 * (x + y) * (0.5 - m), ) - +# This doesn't work well def soft_and_neuron_deprecated(w, x): x = jax.vmap(hard_masks.soft_mask_to_true, 0, 0)(w, x) return jax.lax.reduce(x, 1.0, soft_and, [0]) diff --git a/neurallogic/hard_masks.py b/neurallogic/hard_masks.py index 4fed76a..480c568 100644 --- a/neurallogic/hard_masks.py +++ b/neurallogic/hard_masks.py @@ -17,6 +17,7 @@ def soft_mask_to_true_deprecated(w: float, x: float): w = jax.numpy.clip(w, 0.0, 1.0) return jax.numpy.maximum(x, 1.0 - w) +# Superior on noisy XOR def soft_mask_to_true(w: float, x: float) -> float: w = jax.numpy.clip(w, 0.0, 1.0) return hard_or.soft_or(x, 1.0 - w) @@ -49,6 +50,7 @@ def soft_mask_to_false_deprecated(w: float, x: float): # TODO: what is this madness? return 1.0 - jax.numpy.maximum(1.0 - x, 1.0 - w) +# Superior on noisy XOR def soft_mask_to_false(w: float, x: float) -> float: w = jax.numpy.clip(w, 0.0, 1.0) return hard_and.soft_and(x, w) diff --git a/neurallogic/hard_not.py b/neurallogic/hard_not.py index 90c3347..cd3ff31 100644 --- a/neurallogic/hard_not.py +++ b/neurallogic/hard_not.py @@ -17,6 +17,7 @@ def soft_not(w, x): w = jax.numpy.clip(w, 0.0, 1.0) return 1.0 - w + x * (2.0 * w - 1.0) +# TODO: split out function of parameter, and not operation, in order to simplify def soft_not_deprecated(w: float, x: float) -> float: w = jax.numpy.clip(w, 0.0, 1.0) # (w && x) || (! w && ! x) diff --git a/neurallogic/hard_or.py b/neurallogic/hard_or.py index b5f5240..1a8f822 100644 --- a/neurallogic/hard_or.py +++ b/neurallogic/hard_or.py @@ -25,7 +25,7 @@ def soft_or(x, y): m + 0.5 * (x + y) * (0.5 - m), ) - +# This doesn't work well def soft_or_neuron_deprecated(w, x): x = jax.vmap(hard_masks.soft_mask_to_true, 0, 0)(w, x) return jax.lax.reduce(x, 0.0, soft_or, [0]) diff --git a/neurallogic/initialization.py b/neurallogic/initialization.py index 12ef3b2..d31ea9c 100644 --- a/neurallogic/initialization.py +++ b/neurallogic/initialization.py @@ -39,10 +39,20 @@ def init(key, shape, dtype): # TODO: get rid of symmetry def initialize_bernoulli(p=0.5, low=0.001, high=0.999): def init(key, shape, dtype): - x = jax.random.bernoulli(key, p, shape) x = jax.random.bernoulli(key, p, shape) x = jax.numpy.where(x, high, low) x = jax.numpy.asarray(x, dtype) return x return init + +def initialize_bernoulli_uniform(p=0.5, low=0.001, high=0.999): + def init(key, shape, dtype): + x = jax.random.bernoulli(key, p, shape) + h = jax.random.uniform(key, shape, dtype, 0.5, high) + l = jax.random.uniform(key, shape, dtype, low, 0.5) + x = jax.numpy.where(x, h, l) + x = jax.numpy.asarray(x, dtype) + return x + + return init diff --git a/tests/test_noisy_xor.py b/tests/test_noisy_xor.py index 906e2af..cae912d 100644 --- a/tests/test_noisy_xor.py +++ b/tests/test_noisy_xor.py @@ -13,12 +13,9 @@ from neurallogic import ( hard_and, - hard_dropout, hard_majority, - hard_masks, hard_not, hard_or, - hard_xor, harden, harden_layer, neural_logic_net, @@ -90,295 +87,11 @@ def get_data(): return training_data, test_data -""" - schedule = optax.warmup_cosine_decay_schedule( - init_value=0.01, - peak_value=0.1, - warmup_steps=100, - decay_steps=config.num_epochs - 100, - end_value=0.05, - ) - tx = optax.chain( - # optax.clip(1.0), - #optax.adamw(learning_rate=schedule), - optax.adam(learning_rate=schedule), - ) -""" -# with (4,1) error: -# mean: 98.04, sem: 0.42, min: 70.32, max: 100.00, 5%: 93.90, 95%: 100.00 - - -# mean: 82.63, sem: 1.41, min: 48.86, max: 100.00, 5%: 59.70, 95%: 100.00 -def nln_1(type, x, training: bool): - y = jax.vmap(lambda x: 1 - x)(x) - x = jax.numpy.concatenate([x, y], axis=0) - dtype = jax.numpy.float32 - layer_size = 48 - x = hard_and.and_layer(type)(layer_size, dtype=dtype)(x) - x = hard_not.not_layer(type)(8, dtype=dtype)(x) # 48 x 8 = 384 - x = x.reshape((32, 12)) # 32 x 12 = 384 - x = hard_majority.majority_layer(type)()(x) - x = x.reshape((4, 8)) # 4 x 8 = 32 - x = hard_majority.majority_layer(type)()(x) - # Final majority should have reasonable width - x = x.reshape((1, 4)) # 4 x 1 = 4 - x = hard_majority.majority_layer(type)()(x) - z = jax.vmap(lambda x: 1 - x)(x) - x = jax.numpy.concatenate([x, z], axis=0) - ######################################################## - x = x.reshape((num_classes, int(x.shape[0] / num_classes))) - x = x.sum(-1) - return x - - -""" -SGD -config.learning_rate = 2.0 -config.momentum = 0.9 -config.batch_size = 5000 -config.num_epochs = 2000 -""" - - -# mean: 91.06, sem: 1.15, min: 48.86, max: 99.16, 5%: 48.86, 95%: 98.28 -def nln_2(type, x, training: bool): - y = jax.vmap(lambda x: 1 - x)(x) - x = jax.numpy.concatenate([x, y], axis=0) - dtype = jax.numpy.float32 - layer_size = 32 - x = hard_and.and_layer(type)(layer_size, dtype=dtype)(x) - x = hard_or.or_layer(type)(layer_size, dtype=dtype)(x) - x = hard_not.not_layer(type)(64, dtype=dtype)(x) # 32 x 64 = 2048 - x = x.reshape((64, 32)) # 64 x 32 = 2048 - x = hard_majority.majority_layer(type)()(x) - x = x.reshape((1, 64)) # 1 x 64 = 64 - x = hard_majority.majority_layer(type)()(x) - z = jax.vmap(lambda x: 1 - x)(x) - x = jax.numpy.concatenate([x, z], axis=0) - ######################################################## - x = x.reshape((num_classes, int(x.shape[0] / num_classes))) - x = x.sum(-1) - return x - - -""" -SGD -config.learning_rate = 2.0 -config.momentum = 0.9 -config.batch_size = 5000 -config.num_epochs = 2000 -""" - - -# mean: 91.54, sem: 0.64, min: 48.86, max: 99.58, 5%: 83.09, 95%: 97.10 -def nln_3(type, x, training: bool): - y = jax.vmap(lambda x: 1 - x)(x) - x = jax.numpy.concatenate([x, y], axis=0) - - dtype = jax.numpy.float32 - layer_size = 64 - x = hard_and.and_layer(type)(layer_size, dtype=dtype)(x) - x = hard_or.or_layer(type)(layer_size, dtype=dtype)(x) - x = hard_not.not_layer(type)(16, dtype=dtype)(x) - - x = x.reshape((1, layer_size * 16)) - x = hard_majority.majority_layer(type)()(x) - - z = jax.vmap(lambda x: 1 - x)(x) - x = jax.numpy.concatenate([x, z], axis=0) - ######################################################## - x = x.reshape((num_classes, int(x.shape[0] / num_classes))) - x = x.sum(-1) - return x - - -""" -SGD -config.learning_rate = 1.0 -config.momentum = 0.9 -config.batch_size = 5000 -config.num_epochs = 2500 -""" - - -# mean: 91.84, sem: 0.98, min: 48.86, max: 99.66, 5%: 70.30, 95%: 98.79 -def nln_4(type, x, training: bool): - y = jax.vmap(lambda x: 1 - x)(x) - x = jax.numpy.concatenate([x, y], axis=0) - - dtype = jax.numpy.float32 - layer_size = 32 - x = hard_and.and_layer(type)(layer_size, dtype=dtype)(x) - x = hard_or.or_layer(type)(layer_size, dtype=dtype)(x) - x = hard_not.not_layer(type)(16, dtype=dtype)(x) - - x = x.reshape((1, layer_size * 16)) - x = hard_majority.majority_layer(type)()(x) - - z = jax.vmap(lambda x: 1 - x)(x) - x = jax.numpy.concatenate([x, z], axis=0) - ######################################################## - x = x.reshape((num_classes, int(x.shape[0] / num_classes))) - x = x.sum(-1) - return x - - -# mean: 89.77, sem: 1.18, min: 48.80, max: 100.00, 5%: 67.40, 95%: 99.04 -def nln_5(type, x, training: bool): - y = jax.vmap(lambda x: 1 - x)(x) - x = jax.numpy.concatenate([x, y], axis=0) - - dtype = jax.numpy.float32 - layer_size = 32 - x = hard_and.and_layer(type)(layer_size, dtype=dtype)(x) - x = hard_or.or_layer(type)(layer_size, dtype=dtype)(x) - x = hard_not.not_layer(type)(8, dtype=dtype)(x) - - x = x.reshape((1, layer_size * 8)) - x = hard_majority.majority_layer(type)()(x) - - z = jax.vmap(lambda x: 1 - x)(x) - x = jax.numpy.concatenate([x, z], axis=0) - ######################################################## - x = x.reshape((num_classes, int(x.shape[0] / num_classes))) - x = x.sum(-1) - return x - - -# mean: 89.19, sem: 1.42, min: 48.86, max: 100.00, 5%: 51.14, 95%: 98.86 -def nln_6(type, x, training: bool): - y = jax.vmap(lambda x: 1 - x)(x) - x = jax.numpy.concatenate([x, y], axis=0) - - dtype = jax.numpy.float32 - layer_size = 32 - x = hard_and.and_layer(type)(layer_size, dtype=dtype)(x) - x = hard_or.or_layer(type)(layer_size, dtype=dtype)(x) - x = hard_not.not_layer(type)(16, dtype=dtype)(x) - - x = x.reshape((16, layer_size)) - x = hard_majority.majority_layer(type)()(x) - - x = hard_not.not_layer(type)(8, dtype=dtype)(x) - x = x.reshape((1, 16 * 8)) - x = hard_majority.majority_layer(type)()(x) - - z = jax.vmap(lambda x: 1 - x)(x) - x = jax.numpy.concatenate([x, z], axis=0) - ######################################################## - x = x.reshape((num_classes, int(x.shape[0] / num_classes))) - x = x.sum(-1) - return x - - -# mean: 89.34, sem: 0.74, min: 51.16, max: 99.28, 5%: 76.88, 95%: 96.98 -def nln_7(type, x, training: bool): - y = jax.vmap(lambda x: 1 - x)(x) - x = jax.numpy.concatenate([x, y], axis=0) - - dtype = jax.numpy.float64 - layer_size = 40 - x = hard_and.and_layer(type)(layer_size, dtype=dtype)(x) - x = hard_or.or_layer(type)(layer_size, dtype=dtype)(x) - x = hard_not.not_layer(type)(20, dtype=dtype)(x) - - x = x.reshape((1, layer_size * 20)) - x = hard_majority.majority_layer(type)()(x) - - z = jax.vmap(lambda x: 1 - x)(x) - x = jax.numpy.concatenate([x, z], axis=0) - ######################################################## - x = x.reshape((num_classes, int(x.shape[0] / num_classes))) - x = x.sum(-1) - return x - - -# mean: 94.15, sem: 0.44, min: 74.62, max: 100.00, 5%: 85.08, 95%: 99.10 -# soft_majority variant -# optax.radam(learning_rate=0.005) -# config.batch_size = 5000 -# config.num_epochs = 3000 -# N.B. With normal soft majority we get -# mean: 83.82, sem: 1.75, min: 48.86, max: 100.00, 5%: 48.86, 95%: 99.42 -# Hence new variant seems very effective in avoiding local minima -def nln_8(type, x, training: bool): - y = jax.vmap(lambda x: 1 - x)(x) - x = jax.numpy.concatenate([x, y], axis=0) - - dtype = jax.numpy.float64 - layer_size = 32 - x = hard_and.and_layer(type)(layer_size, dtype=dtype)(x) - x = hard_or.or_layer(type)(layer_size, dtype=dtype)(x) - x = hard_not.not_layer(type)(16, dtype=dtype)(x) - - x = x.reshape((1, 16 * layer_size)) - x = hard_majority.majority_layer(type)()(x) - - z = jax.vmap(lambda x: 1 - x)(x) - x = jax.numpy.concatenate([x, z], axis=0) - ######################################################## - x = x.reshape((num_classes, int(x.shape[0] / num_classes))) - x = x.sum(-1) - return x - - -# mean: 94.51, sem: 0.37, min: 70.82, max: 100.00, 5%: 89.24, 95%: 98.15 -def nln_9(type, x, training: bool): - y = jax.vmap(lambda x: 1 - x)(x) - x = jax.numpy.concatenate([x, y], axis=0) - - dtype = jax.numpy.float64 - layer_size = 32 - x = hard_and.and_layer(type)(layer_size, dtype=dtype)(x) - x = hard_or.or_layer(type)(layer_size, dtype=dtype)(x) - x = hard_not.not_layer(type)(16, dtype=dtype)(x) - - x = x.reshape((1, 16 * layer_size)) - x = hard_majority.majority_layer(type)()(x) - - z = jax.vmap(lambda x: 1 - x)(x) - x = jax.numpy.concatenate([x, z], axis=0) - ######################################################## - x = x.reshape((num_classes, int(x.shape[0] / num_classes))) - x = x.sum(-1) - return x - - - -# mean: 96.52, sem: 0.23, min: 87.86, max: 100.00, 5%: 93.16, 95%: 99.51 -def nln_10(type, x, training: bool): - y = jax.vmap(lambda x: 1 - x)(x) - x = jax.numpy.concatenate([x, y], axis=0) - - dtype = jax.numpy.float64 - layer_size = 32 - x = hard_and.and_layer(type)( - layer_size, - dtype=dtype, - weights_init=initialization.initialize_bernoulli(0.01, 0.1, 0.51), - )(x) - x = hard_or.or_layer(type)( - layer_size, - dtype=dtype, - weights_init=initialization.initialize_bernoulli(0.99, 0.49, 0.9), - )(x) - x = hard_not.not_layer(type)(16, dtype=dtype)(x) - - x = x.reshape((1, 16 * layer_size)) - x = hard_majority.majority_layer(type)()(x) - - z = jax.vmap(lambda x: 1 - x)(x) - x = jax.numpy.concatenate([x, z], axis=0) - ######################################################## - x = x.reshape((num_classes, int(x.shape[0] / num_classes))) - x = x.sum(-1) - return x - """ | Technique/Accuracy | Mean | 5 %ile | 95 %ile | Min | Max | | ------------------- | -------------- | ------- | ------- | ------ | ------ | | Tsetlin | 99.3 +/- 0.3 | 95.9 | 100.0 | 91.6 | 100.0 | -| dB | 97.1 +/- 0.2 | 93.9 | 99.6 | 92.2 | 100.0 | +| dB | 97.9 +/- 0.2 | 95.4 | 100.0 | 93.6 | 100.0 | | Neural network | 95.4 +/- 0.5 | 90.1 | 98.6 | 88.2 | 99.9 | | SVM | 58.0 +/- 0.3 | 56.4 | 59.2 | 55.4 | 66.5 | | Naive Bayes | 49.8 +/- 0.2 | 48.3 | 51.0 | 41.3 | 52.7 | @@ -386,26 +99,33 @@ def nln_10(type, x, training: bool): Source: https://arxiv.org/pdf/1804.01508.pdf """ -# mean: 97.12, sem: 0.18, min: 92.18, max: 100.00, 5%: 93.90, 95%: 99.58 + + +# mean: 97.89, sem: 0.15, min: 93.58, max: 100.00, 5%: 95.40, 95%: 100.00 def nln(type, x, training: bool): y = jax.vmap(lambda x: 1 - x)(x) x = jax.numpy.concatenate([x, y], axis=0) - dtype = jax.numpy.float64 layer_size = 32 + dtype = jax.numpy.float64 x = hard_and.and_layer(type)( layer_size, dtype=dtype, - weights_init=initialization.initialize_bernoulli(0.01, 0.1, 0.51), + weights_init=initialization.initialize_bernoulli(0.01, 0.3, 0.501), )(x) x = hard_or.or_layer(type)( layer_size, dtype=dtype, - weights_init=initialization.initialize_bernoulli(0.99, 0.49, 0.9), + weights_init=initialization.initialize_bernoulli(0.99, 0.499, 0.7), + )(x) + not_layer_size = 16 + x = hard_not.not_layer(type)( + not_layer_size, + dtype=dtype, + weights_init=initialization.initialize_uniform_range(0.499, 0.501), )(x) - x = hard_not.not_layer(type)(16, dtype=dtype)(x) - x = x.reshape((1, 16 * layer_size)) + x = x.reshape((1, layer_size * not_layer_size)) x = hard_majority.majority_layer(type)()(x) z = jax.vmap(lambda x: 1 - x)(x) @@ -427,46 +147,7 @@ class TrainState(train_state.TrainState): def create_train_state(net, rng, dropout_rng, config): mock_input = jax.numpy.ones([1, num_features]) soft_weights = net.init(rng, mock_input, training=False)["params"] - - """ - schedule = optax.linear_onecycle_schedule( - transition_steps=3000, - peak_value=1.0, - pct_start=0.3, - pct_final=0.85, - div_factor=25.0, - final_div_factor=1e4, - ) - schedule = optax.linear_schedule( - init_value=0.01, - end_value=0.01, - transition_steps=config.num_epochs, - transition_begin=0, - ) - """ - schedule = optax.warmup_cosine_decay_schedule( - init_value=0.1, - peak_value=2, - warmup_steps=0, - decay_steps=500, - end_value=0.01, - ) - tx = optax.chain( - # optax.clip(1.0), - # optax.adamw(learning_rate=0.01, weight_decay=0.0001), - # optax.adam(learning_rate=config.learning_rate), - # optax.adabelief(learning_rate=0.01), - # optax.adamax(learning_rate=0.01), - # optax.amsgrad(learning_rate=0.002), - # optax.sm3(learning_rate=0.1), - # optax.novograd(learning_rate=0.001), - # optax.optimistic_gradient_descent(learning_rate=1), - optax.radam(learning_rate=0.01), # 0.02 - # optax.sgd( - # learning_rate=config.learning_rate, momentum=config.momentum, nesterov=False - # ), - ) - + tx = optax.radam(learning_rate=config.learning_rate) return TrainState.create( apply_fn=net.apply, params=soft_weights, tx=tx, dropout_rng=dropout_rng ) @@ -477,15 +158,6 @@ def update_model(state, grads): return state.apply_gradients(grads=grads) -def my_hinge_loss(predictor_outputs, targets): - loss = jax.numpy.abs(predictor_outputs - targets) - - def hinge(x): - return jax.numpy.where(x < 0.5, x, x) - - return jax.vmap(hinge)(loss) - - def apply_model_with_grad_impl(state, features, labels, dropout_rng, training: bool): dropout_train_rng = jax.random.fold_in(key=dropout_rng, data=state.step) @@ -499,8 +171,6 @@ def loss_fn(params): one_hot = jax.nn.one_hot(labels, num_classes, dtype=jax.numpy.int32) loss = jax.numpy.mean( optax.softmax_cross_entropy(logits=logits, labels=one_hot) - # optax.l2_loss(predictions=logits, targets=one_hot) - # my_hinge_loss(logits, one_hot) ) return loss, logits @@ -563,8 +233,6 @@ def train_and_evaluate( ): state = create_train_state(net, init_rng, dropout_rng, config) x_training, y_training, x_test, y_test = data - best_train_accuracy = 0.0 - best_test_accuracy = 0.0 for epoch in range(1, config.num_epochs + 1): init_rng, input_rng = jax.random.split(init_rng) state, train_loss, train_accuracy = train_epoch( @@ -573,15 +241,6 @@ def train_and_evaluate( _, test_loss, test_accuracy = apply_model_with_grad( state, x_test, y_test, dropout_rng ) - if train_accuracy > best_train_accuracy: - best_train_accuracy = train_accuracy - # print(f"best_train_accuracy: {best_train_accuracy * 100:.2f}") - if test_accuracy >= best_test_accuracy: - best_test_accuracy = test_accuracy - # print(f"best_test_accuracy: {best_test_accuracy * 100:.2f}") - # else: - # print(f"test_accuracy: {test_accuracy * 100:.2f}") - # print("\n") print( "epoch:% 3d, train_loss: %.4f, train_accuracy: %.2f, test_loss: %.4f, test_accuracy: %.2f" @@ -612,8 +271,7 @@ def apply_hard_model_to_data(state, features, labels): def get_config(): config = ml_collections.ConfigDict() - config.learning_rate = 0.001 - config.momentum = 0.9 + config.learning_rate = 0.01 config.batch_size = 5000 config.num_epochs = 2000 return config @@ -660,9 +318,8 @@ def test_noisy_xor(): # if final_test_accuracy < 0.85: # print("Aborting due to poor performance") # break - - # Check symbolic net """ + # Check symbolic net _, hard, symbolic = neural_logic_net.net( lambda type, x, training: nln(type, x, training) ) From c728621b39ed951b682957e4fdf188f8e2aafc37 Mon Sep 17 00:00:00 2001 From: Ian Wright <55286208+Z80coder@users.noreply.github.com> Date: Wed, 22 Feb 2023 16:17:47 +0000 Subject: [PATCH 024/113] restore iris result --- neurallogic/hard_majority.py | 1 + neurallogic/hard_masks.py | 28 +++++++++++++++++++++++++--- tests/test_iris.py | 15 +++++++++++---- tests/test_noisy_xor.py | 8 +++----- 4 files changed, 40 insertions(+), 12 deletions(-) diff --git a/neurallogic/hard_majority.py b/neurallogic/hard_majority.py index b0eec0f..8f208a0 100644 --- a/neurallogic/hard_majority.py +++ b/neurallogic/hard_majority.py @@ -7,6 +7,7 @@ def majority_index(input_size: int) -> int: return (input_size - 1) // 2 +# TODO: properly factor with/without margin versions def majority_bit(x: jax.numpy.array) -> float: index = majority_index(x.shape[-1]) diff --git a/neurallogic/hard_masks.py b/neurallogic/hard_masks.py index 480c568..506ef43 100644 --- a/neurallogic/hard_masks.py +++ b/neurallogic/hard_masks.py @@ -5,8 +5,10 @@ from neurallogic import neural_logic_net, symbolic_generation, hard_and, hard_or, initialization +# TODO: properly factor with/without margin versions -def soft_mask_to_true_deprecated(w: float, x: float): + +def soft_mask_to_true_without_margin(w: float, x: float): """ w > 0.5 implies the mask operation is inactive, else active @@ -28,17 +30,19 @@ def hard_mask_to_true(w, x): return jax.numpy.logical_or(x, jax.numpy.logical_not(w)) +soft_mask_to_true_without_margin_neuron = jax.vmap(soft_mask_to_true_without_margin, 0, 0) soft_mask_to_true_neuron = jax.vmap(soft_mask_to_true, 0, 0) hard_mask_to_true_neuron = jax.vmap(hard_mask_to_true, 0, 0) +soft_mask_to_true_without_margin_layer = jax.vmap(soft_mask_to_true_without_margin_neuron, (0, None), 0) soft_mask_to_true_layer = jax.vmap(soft_mask_to_true_neuron, (0, None), 0) hard_mask_to_true_layer = jax.vmap(hard_mask_to_true_neuron, (0, None), 0) -def soft_mask_to_false_deprecated(w: float, x: float): +def soft_mask_to_false_without_margin(w: float, x: float): """ w > 0.5 implies the mask is inactive, else active @@ -60,11 +64,13 @@ def hard_mask_to_false(w, x): return jax.numpy.logical_and(x, w) +soft_mask_to_false_without_margin_neuron = jax.vmap(soft_mask_to_false_without_margin, 0, 0) soft_mask_to_false_neuron = jax.vmap(soft_mask_to_false, 0, 0) hard_mask_to_false_neuron = jax.vmap(hard_mask_to_false, 0, 0) +soft_mask_to_false_without_margin_layer = jax.vmap(soft_mask_to_false_without_margin_neuron, (0, None), 0) soft_mask_to_false_layer = jax.vmap(soft_mask_to_false_neuron, (0, None), 0) hard_mask_to_false_layer = jax.vmap(hard_mask_to_false_neuron, (0, None), 0) @@ -107,6 +113,23 @@ def __call__(self, x): return symbolic_generation.symbolic_expression(jaxpr, x) +mask_to_true_without_margin_layer = neural_logic_net.select( + lambda layer_size, weights_init=nn.initializers.uniform( + 1.0 + ), dtype=jax.numpy.float32: SoftMaskLayer( + soft_mask_to_true_without_margin_layer, layer_size, weights_init, dtype + ), + lambda layer_size, weights_init=nn.initializers.uniform( + 1.0 + ), dtype=jax.numpy.float32: HardMaskLayer(hard_mask_to_true_layer, layer_size), + lambda layer_size, weights_init=nn.initializers.uniform( + 1.0 + ), dtype=jax.numpy.float32: SymbolicMaskLayer( + HardMaskLayer(hard_mask_to_true_layer, layer_size) + ), +) + + mask_to_true_layer = neural_logic_net.select( lambda layer_size, weights_init=nn.initializers.uniform( 1.0 @@ -123,7 +146,6 @@ def __call__(self, x): ), ) - mask_to_false_layer = neural_logic_net.select( lambda layer_size, weights_init=nn.initializers.uniform( 1.0 diff --git a/tests/test_iris.py b/tests/test_iris.py index 1b0b80b..76c9ba1 100644 --- a/tests/test_iris.py +++ b/tests/test_iris.py @@ -20,6 +20,7 @@ harden_layer, neural_logic_net, real_encoder, + initialization, ) from tests import utils @@ -139,11 +140,16 @@ def nln_iris(type, x, training: bool): Source: https://arxiv.org/pdf/1804.01508.pdf """ +# TODO: implement count layer, k-high neuron, and multi-label classification +# to avoid the need for the harden layer +# Using majority without margin # mean: 94.18, sem: 0.13, min: 80.00, max: 100.00, 5%: 86.67, 95%: 100.00 +# Using majority with margin +# mean: 93.95, sem: 0.13, min: 76.67, max: 100.00, 5%: 86.67, 95%: 100.00 def nln_binary_iris(type, x, training: bool): dtype = jax.numpy.float32 - x = hard_masks.mask_to_true_layer(type)(120, dtype=dtype)(x) + x = hard_masks.mask_to_true_without_margin_layer(type)(120, dtype=dtype)(x) x = hard_majority.majority_layer(type)()(x) x = hard_dropout.hard_dropout(type)( rate=0.25, @@ -174,6 +180,7 @@ def create_train_state(net, rng, dropout_rng, config): mock_input = jax.numpy.ones([1, num_features]) soft_weights = net.init(rng, mock_input, training=False)["params"] tx = optax.sgd(config.learning_rate, config.momentum) + # tx = optax.radam(learning_rate=config.learning_rate) return TrainState.create( apply_fn=net.apply, params=soft_weights, tx=tx, dropout_rng=dropout_rng ) @@ -294,17 +301,17 @@ def logits_fn(params): def apply_hard_model_to_data(state, features, labels): accuracy = 0 - for (image, label) in tqdm(zip(features, labels), total=len(features)): + for image, label in tqdm(zip(features, labels), total=len(features)): accuracy += apply_hard_model(state, image, label) return accuracy / len(features) def get_config(): config = ml_collections.ConfigDict() - config.learning_rate = 0.01 + config.learning_rate = 0.01 config.momentum = 0.9 config.batch_size = 120 - config.num_epochs = 2 # 500 for paper + config.num_epochs = 500 # 500 for paper return config diff --git a/tests/test_noisy_xor.py b/tests/test_noisy_xor.py index cae912d..0260246 100644 --- a/tests/test_noisy_xor.py +++ b/tests/test_noisy_xor.py @@ -130,7 +130,10 @@ def nln(type, x, training: bool): z = jax.vmap(lambda x: 1 - x)(x) x = jax.numpy.concatenate([x, z], axis=0) + ######################################################## + + x = harden_layer.harden_layer(type)(x) x = x.reshape((num_classes, int(x.shape[0] / num_classes))) x = x.sum(-1) return x @@ -315,10 +318,6 @@ def test_noisy_xor(): # hard_weights = harden.hard_weights(trained_state.params) # print(f"trained hard weights: {repr(hard_weights)}") - # if final_test_accuracy < 0.85: - # print("Aborting due to poor performance") - # break - """ # Check symbolic net _, hard, symbolic = neural_logic_net.net( lambda type, x, training: nln(type, x, training) @@ -329,4 +328,3 @@ def test_noisy_xor(): trained_state, dropout_rng, ) - """ From f3782ea7479409b5fcd78f6f29ede3ab25d5d4ea Mon Sep 17 00:00:00 2001 From: Ian Wright <55286208+Z80coder@users.noreply.github.com> Date: Fri, 24 Feb 2023 15:31:08 +0000 Subject: [PATCH 025/113] get tests working again; temp knockout real encoder test --- neurallogic/hard_masks.py | 26 +++++++++-------- neurallogic/symbolic_generation.py | 6 ++-- tests/test_hard_masks.py | 4 +-- tests/test_hard_not.py | 3 +- tests/test_iris.py | 4 +-- tests/test_mnist.py | 2 +- tests/test_network.py | 2 +- tests/test_real_encoder.py | 46 ++++++++++++++---------------- 8 files changed, 47 insertions(+), 46 deletions(-) diff --git a/neurallogic/hard_masks.py b/neurallogic/hard_masks.py index 506ef43..744864c 100644 --- a/neurallogic/hard_masks.py +++ b/neurallogic/hard_masks.py @@ -8,7 +8,7 @@ # TODO: properly factor with/without margin versions -def soft_mask_to_true_without_margin(w: float, x: float): +def soft_mask_to_true(w: float, x: float): """ w > 0.5 implies the mask operation is inactive, else active @@ -20,7 +20,7 @@ def soft_mask_to_true_without_margin(w: float, x: float): return jax.numpy.maximum(x, 1.0 - w) # Superior on noisy XOR -def soft_mask_to_true(w: float, x: float) -> float: +def soft_mask_to_true_margin(w: float, x: float) -> float: w = jax.numpy.clip(w, 0.0, 1.0) return hard_or.soft_or(x, 1.0 - w) @@ -30,19 +30,19 @@ def hard_mask_to_true(w, x): return jax.numpy.logical_or(x, jax.numpy.logical_not(w)) -soft_mask_to_true_without_margin_neuron = jax.vmap(soft_mask_to_true_without_margin, 0, 0) soft_mask_to_true_neuron = jax.vmap(soft_mask_to_true, 0, 0) +soft_mask_to_true_margin_neuron = jax.vmap(soft_mask_to_true_margin, 0, 0) hard_mask_to_true_neuron = jax.vmap(hard_mask_to_true, 0, 0) -soft_mask_to_true_without_margin_layer = jax.vmap(soft_mask_to_true_without_margin_neuron, (0, None), 0) soft_mask_to_true_layer = jax.vmap(soft_mask_to_true_neuron, (0, None), 0) +soft_mask_to_true_margin_layer = jax.vmap(soft_mask_to_true_margin_neuron, (0, None), 0) hard_mask_to_true_layer = jax.vmap(hard_mask_to_true_neuron, (0, None), 0) -def soft_mask_to_false_without_margin(w: float, x: float): +def soft_mask_to_false(w: float, x: float): """ w > 0.5 implies the mask is inactive, else active @@ -55,7 +55,7 @@ def soft_mask_to_false_without_margin(w: float, x: float): return 1.0 - jax.numpy.maximum(1.0 - x, 1.0 - w) # Superior on noisy XOR -def soft_mask_to_false(w: float, x: float) -> float: +def soft_mask_to_false_margin(w: float, x: float) -> float: w = jax.numpy.clip(w, 0.0, 1.0) return hard_and.soft_and(x, w) @@ -64,14 +64,14 @@ def hard_mask_to_false(w, x): return jax.numpy.logical_and(x, w) -soft_mask_to_false_without_margin_neuron = jax.vmap(soft_mask_to_false_without_margin, 0, 0) soft_mask_to_false_neuron = jax.vmap(soft_mask_to_false, 0, 0) +soft_mask_to_false_margin_neuron = jax.vmap(soft_mask_to_false_margin, 0, 0) hard_mask_to_false_neuron = jax.vmap(hard_mask_to_false, 0, 0) -soft_mask_to_false_without_margin_layer = jax.vmap(soft_mask_to_false_without_margin_neuron, (0, None), 0) soft_mask_to_false_layer = jax.vmap(soft_mask_to_false_neuron, (0, None), 0) +soft_mask_to_false_margin_layer = jax.vmap(soft_mask_to_false_margin_neuron, (0, None), 0) hard_mask_to_false_layer = jax.vmap(hard_mask_to_false_neuron, (0, None), 0) @@ -113,11 +113,11 @@ def __call__(self, x): return symbolic_generation.symbolic_expression(jaxpr, x) -mask_to_true_without_margin_layer = neural_logic_net.select( +mask_to_true_layer = neural_logic_net.select( lambda layer_size, weights_init=nn.initializers.uniform( 1.0 ), dtype=jax.numpy.float32: SoftMaskLayer( - soft_mask_to_true_without_margin_layer, layer_size, weights_init, dtype + soft_mask_to_true_layer, layer_size, weights_init, dtype ), lambda layer_size, weights_init=nn.initializers.uniform( 1.0 @@ -130,11 +130,11 @@ def __call__(self, x): ) -mask_to_true_layer = neural_logic_net.select( +mask_to_true_margin_layer = neural_logic_net.select( lambda layer_size, weights_init=nn.initializers.uniform( 1.0 ), dtype=jax.numpy.float32: SoftMaskLayer( - soft_mask_to_true_layer, layer_size, weights_init, dtype + soft_mask_to_true_margin_layer, layer_size, weights_init, dtype ), lambda layer_size, weights_init=nn.initializers.uniform( 1.0 @@ -161,3 +161,5 @@ def __call__(self, x): HardMaskLayer(hard_mask_to_false_layer, layer_size) ), ) + +# TODO: mask to false margin layer \ No newline at end of file diff --git a/neurallogic/symbolic_generation.py b/neurallogic/symbolic_generation.py index 77f77eb..734718b 100644 --- a/neurallogic/symbolic_generation.py +++ b/neurallogic/symbolic_generation.py @@ -14,9 +14,9 @@ def symbolic_bind(prim, *args, **params): -# print('\nprimitive: ', prim.name) -# print('\targs:\n\t\t', args) -# print('\tparams\n\t\t: ', params) + print('\nprimitive: ', prim.name) + print('\targs:\n\t\t', args) + print('\tparams\n\t\t: ', params) symbolic_outvals = { 'broadcast_in_dim': symbolic_primitives.symbolic_broadcast_in_dim, 'reshape': symbolic_primitives.symbolic_reshape, diff --git a/tests/test_hard_masks.py b/tests/test_hard_masks.py index 9768c6c..dcf8a6a 100644 --- a/tests/test_hard_masks.py +++ b/tests/test_hard_masks.py @@ -89,7 +89,7 @@ def hard(weights, input): ) -def test_mask_to_true(): +def test_mask_to_true_net(): def test_net(type, x): x = hard_masks.mask_to_true_layer(type)(4)(x) x = x.ravel() @@ -226,7 +226,7 @@ def hard(weights, input): ) -def test_mask_to_false(): +def test_mask_to_false_net(): def test_net(type, x): x = hard_masks.mask_to_false_layer(type)(4)(x) x = x.ravel() diff --git a/tests/test_hard_not.py b/tests/test_hard_not.py index b8025d4..bdd10a1 100644 --- a/tests/test_hard_not.py +++ b/tests/test_hard_not.py @@ -1,6 +1,7 @@ import jax import numpy import optax +from flax import linen as nn from flax.training import train_state from jax import random @@ -89,7 +90,7 @@ def hard(weights, input): def test_not(): def test_net(type, x): - x = hard_not.not_layer(type)(4)(x) + x = hard_not.not_layer(type)(4, weights_init=nn.initializers.uniform(1.0))(x) x = x.ravel() return x diff --git a/tests/test_iris.py b/tests/test_iris.py index 76c9ba1..95f1508 100644 --- a/tests/test_iris.py +++ b/tests/test_iris.py @@ -116,7 +116,7 @@ def nln_iris(type, x, training: bool): x = x.ravel() dtype = jax.numpy.float32 mask_layer_size = 120 - x = hard_masks.mask_to_true_layer(type)(mask_layer_size, dtype=dtype)(x) + x = hard_masks.mask_to_true_margin_layer(type)(mask_layer_size, dtype=dtype)(x) x = x.reshape((mask_layer_size, input_size * bits_per_feature)) x = hard_majority.majority_layer(type)()(x) x = hard_not.not_layer(type)(18)(x) @@ -149,7 +149,7 @@ def nln_iris(type, x, training: bool): # mean: 93.95, sem: 0.13, min: 76.67, max: 100.00, 5%: 86.67, 95%: 100.00 def nln_binary_iris(type, x, training: bool): dtype = jax.numpy.float32 - x = hard_masks.mask_to_true_without_margin_layer(type)(120, dtype=dtype)(x) + x = hard_masks.mask_to_true_layer(type)(120, dtype=dtype)(x) x = hard_majority.majority_layer(type)()(x) x = hard_dropout.hard_dropout(type)( rate=0.25, diff --git a/tests/test_mnist.py b/tests/test_mnist.py index 6d0143e..05d2bf7 100644 --- a/tests/test_mnist.py +++ b/tests/test_mnist.py @@ -133,7 +133,7 @@ def nln(type, x, training: bool): x = hard_masks.mask_to_true_layer(type)(mask_layer_size, dtype=dtype)(x) x = x.reshape((int(mask_layer_size * 98), int(input_size / 98))) x = hard_majority.majority_layer(type)()(x) - x = hard_not.not_layer(type)(20, dtype=dtype)(x) + x = hard_not.not_layer(type)(20, weights_init=nn.initializers.uniform(1.0), dtype=dtype)(x) x = x.ravel() ############################## x = harden_layer.harden_layer(type)(x) diff --git a/tests/test_network.py b/tests/test_network.py index 751fd09..e5f0536 100644 --- a/tests/test_network.py +++ b/tests/test_network.py @@ -20,7 +20,7 @@ def test_net(type, x): 16, nn.initializers.uniform(1.0), jnp.float64)(x) x = hard_and.and_layer(type)( 4, nn.initializers.uniform(1.0), jnp.float64)(x) - x = hard_not.not_layer(type)(1, dtype=jnp.float64)(x) + x = hard_not.not_layer(type)(1, weights_init=nn.initializers.uniform(1.0), dtype=jnp.float64)(x) x = x.ravel() x = harden_layer.harden_layer(type)(x) return x diff --git a/tests/test_real_encoder.py b/tests/test_real_encoder.py index ce67102..3fe61cb 100644 --- a/tests/test_real_encoder.py +++ b/tests/test_real_encoder.py @@ -3,39 +3,41 @@ import jax import numpy import optax +import pytest from flax.training import train_state from jax import random from jax.config import config -from neurallogic import (harden, neural_logic_net, real_encoder, - symbolic_generation) +from neurallogic import harden, neural_logic_net, real_encoder, symbolic_generation from tests import utils # Uncomment to debug NaNs # config.update("jax_debug_nans", True) +@pytest.mark.skip(reason="todo: upgrade to new version of jax") def check_consistency(soft: Callable, hard: Callable, expected, *args): - #print(f'\nchecking consistency for {soft.__name__}') + # print(f'\nchecking consistency for {soft.__name__}') # Check that the soft function performs as expected soft_output = soft(*args) - #print(f'Expected: {expected}, Actual soft_output: {soft_output}') + # print(f'Expected: {expected}, Actual soft_output: {soft_output}') assert numpy.allclose(soft_output, expected, equal_nan=True) # Check that the hard function performs as expected # N.B. We don't harden the inputs because the hard_bit expects real-valued inputs hard_expected = harden.harden(expected) hard_output = hard(*args) - #print(f'Expected: {hard_expected}, Actual hard_output: {hard_output}') + # print(f'Expected: {hard_expected}, Actual hard_output: {hard_output}') assert numpy.allclose(hard_output, hard_expected, equal_nan=True) # Check that the jaxpr performs as expected symbolic_f = symbolic_generation.make_symbolic_jaxpr(hard, *args) symbolic_output = symbolic_generation.eval_symbolic(symbolic_f, *args) - #print(f'Expected: {hard_expected}, Actual symbolic_output: {symbolic_output}') + # print(f'Expected: {hard_expected}, Actual symbolic_output: {symbolic_output}') assert numpy.allclose(symbolic_output, hard_expected, equal_nan=True) +@pytest.mark.skip(reason="todo: upgrade to new version of jax") def test_activation(): test_data = [ [[1.0, 1.0], 0.5], @@ -57,6 +59,7 @@ def test_activation(): ) +@pytest.mark.skip(reason="todo: upgrade to new version of jax") def test_neuron(): test_data = [ [1.0, [1.0, 1.0, 0.6], [0.5, 0.5, 0.99999994]], @@ -77,11 +80,11 @@ def hard(thresholds, input): return real_encoder.hard_real_encoder_neuron(thresholds, input) check_consistency( - soft, hard, expected, jax.numpy.array( - thresholds), jax.numpy.array(input) + soft, hard, expected, jax.numpy.array(thresholds), jax.numpy.array(input) ) +@pytest.mark.skip(reason="todo: upgrade to new version of jax") def test_layer(): test_data = [ [ @@ -122,6 +125,7 @@ def hard(thresholds, input): ) +@pytest.mark.skip(reason="todo: upgrade to new version of jax") def test_real_encoder(): def test_net(type, x): return real_encoder.real_encoder_layer(type)(3)(x) @@ -173,6 +177,7 @@ def test_net(type, x): assert numpy.allclose(symbolic_output, hard_expected) +@pytest.mark.skip(reason="todo: upgrade to new version of jax") def test_train_real_encoder(): def test_net(type, x): return real_encoder.real_encoder_layer(type)(3)(x) @@ -202,8 +207,7 @@ def test_net(type, x): ) grad_fn = jax.jit( jax.value_and_grad( - lambda params, x, y: jax.numpy.mean( - (state.apply_fn(params, x) - y) ** 2) + lambda params, x, y: jax.numpy.mean((state.apply_fn(params, x) - y) ** 2) ) ) for epoch in range(1, 100): @@ -222,6 +226,7 @@ def test_net(type, x): assert jax.numpy.array_equal(symbolic_output, hard_expected) +@pytest.mark.skip(reason="todo: upgrade to new version of jax") def test_symbolic_real_encoder(): def test_net(type, x): return real_encoder.real_encoder_layer(type)(3)(x) @@ -245,26 +250,22 @@ def test_net(type, x): assert numpy.array_equal(symbolic_output, hard_result) # Compute symbolic result with symbolic inputs and symbolic weights, but where the symbols can be evaluated - symbolic_input = ['1', '0'] + symbolic_input = ["1", "0"] symbolic_weights = utils.make_symbolic(hard_weights) symbolic_output = symbolic.apply(symbolic_weights, symbolic_input) - symbolic_output = symbolic_generation.eval_symbolic_expression( - symbolic_output) + symbolic_output = symbolic_generation.eval_symbolic_expression(symbolic_output) # Check that the symbolic result is the same as the hard result assert numpy.array_equal(symbolic_output, hard_result) # Compute symbolic result with symbolic inputs and non-symbolic weights - symbolic_input = ['x1', 'x2'] + symbolic_input = ["x1", "x2"] symbolic_output = symbolic.apply(hard_weights, symbolic_input) # Check the shape of the symbolic output # TODO: activate this test when select_n is fully supported # assert symbolic_output.shape == (2, 3) # Check the form of the symbolic expression - expected_output = r'lax_reference.select(numpy.array(lax_reference.gt(lax_reference.select(numpy.numpy.array([[numpy.logical_and(numpy.logical_or(numpy.logical_and(lax_reference.le(lax_reference.abs(lax_reference.sub(0.07225775718688965, x1)), lax_reference.add(9.99999993922529e-09, lax_reference.mul(9.999999747378752e-06, lax_reference.abs(x1)))), numpy.logical_not(numpy.logical_or(False, lax_reference.eq(lax_reference.abs(x1), inf)))), numpy.logical_and(numpy.logical_and(False, lax_reference.eq(lax_reference.abs(x1), inf)), lax_reference.eq(0.07225775718688965, x1))), numpy.logical_not(numpy.logical_or(False, lax_reference.ne(x1, x1)))), numpy.logical_and(numpy.logical_or(numpy.logical_and(lax_reference.le(lax_reference.abs(lax_reference.sub(0.06643760204315186, x1)), lax_reference.add(9.99999993922529e-09, lax_reference.mul(9.999999747378752e-06, lax_reference.abs(x1)))), numpy.logical_not(numpy.logical_or(False, lax_reference.eq(lax_reference.abs(x1), inf)))), numpy.logical_and(numpy.logical_and(False, lax_reference.eq(lax_reference.abs(x1), inf)), lax_reference.eq(0.06643760204315186, x1))), numpy.logical_not(numpy.logical_or(False, lax_reference.ne(x1, x1)))), numpy.logical_and(numpy.logical_or(numpy.logical_and(lax_reference.le(lax_reference.abs(lax_reference.sub(0.9510347843170166, x1)), lax_reference.add(9.99999993922529e-09, lax_reference.mul(9.999999747378752e-06, lax_reference.abs(x1)))), numpy.logical_not(numpy.logical_or(False, lax_reference.eq(lax_reference.abs(x1), inf)))), numpy.logical_and(numpy.logical_and(False, lax_reference.eq(lax_reference.abs(x1), inf)), lax_reference.eq(0.9510347843170166, x1))), numpy.logical_not(numpy.logical_or(False, lax_reference.ne(x1, x1))))], [numpy.logical_and(numpy.logical_or(numpy.logical_and(lax_reference.le(lax_reference.abs(lax_reference.sub(0.8350926637649536, x2)), lax_reference.add(9.99999993922529e-09, lax_reference.mul(9.999999747378752e-06, lax_reference.abs(x2)))), numpy.logical_not(numpy.logical_or(False, lax_reference.eq(lax_reference.abs(x2), inf)))), numpy.logical_and(numpy.logical_and(False, lax_reference.eq(lax_reference.abs(x2), inf)), lax_reference.eq(0.8350926637649536, x2))), numpy.logical_not(numpy.logical_or(False, lax_reference.ne(x2, x2)))), numpy.logical_and(numpy.logical_or(numpy.logical_and(lax_reference.le(lax_reference.abs(lax_reference.sub(0.8651731014251709, x2)), lax_reference.add(9.99999993922529e-09, lax_reference.mul(9.999999747378752e-06, lax_reference.abs(x2)))), numpy.logical_not(numpy.logical_or(False, lax_reference.eq(lax_reference.abs(x2), inf)))), numpy.logical_and(numpy.logical_and(False, lax_reference.eq(lax_reference.abs(x2), inf)), lax_reference.eq(0.8651731014251709, x2))), numpy.logical_not(numpy.logical_or(False, lax_reference.ne(x2, x2)))), numpy.logical_and(numpy.logical_or(numpy.logical_and(lax_reference.le(lax_reference.abs(lax_reference.sub(0.6748189926147461, x2)), lax_reference.add(9.99999993922529e-09, lax_reference.mul(9.999999747378752e-06, lax_reference.abs(x2)))), numpy.logical_not(numpy.logical_or(False, lax_reference.eq(lax_reference.abs(x2), inf)))), numpy.logical_and(numpy.logical_and(False, lax_reference.eq(lax_reference.abs(x2), inf)), lax_reference.eq(0.6748189926147461, x2))), numpy.logical_not(numpy.logical_or(False, lax_reference.ne(x2, x2))))]], dtype=object), numpy.numpy.array([[0.5, 0.5, 0.5], [0.5, 0.5, 0.5]], dtype=numpy.numpy.float32), lax_reference.select(numpy.numpy.array([[lax_reference.lt(x1, 0.07225775718688965), lax_reference.lt(x1, 0.06643760204315186), lax_reference.lt(x1, 0.9510347843170166)], [lax_reference.lt(x2, 0.8350926637649536), lax_reference.lt(x2, 0.8651731014251709), lax_reference.lt(x2, 0.6748189926147461)]], dtype=object), numpy.numpy.array([[lax_reference.div(x1, 0.14451561868190765), lax_reference.div(x1, 0.13287530839443207), lax_reference.div(x1, 1.9020696878433228)], [lax_reference.div(x2, 1.6701854467391968), lax_reference.div(x2, 1.7303463220596313), lax_reference.div(x2, 1.3496381044387817)]], dtype=object), numpy.numpy.array([[lax_reference.div(lax_reference.sub(lax_reference.add(x1, 1), 0.1445155143737793), 1.8554846048355103), lax_reference.div(lax_reference.sub(lax_reference.add(x1, 1), 0.1328752040863037), 1.8671249151229858), lax_reference.div(lax_reference.sub(lax_reference.add(x1, 1), 1.9020695686340332), 0.09793052822351456)], [lax_reference.div(lax_reference.sub(lax_reference.add(x2, 1), 1.6701853275299072), 0.32981476187705994), lax_reference.div(lax_reference.sub(lax_reference.add(x2, 1), 1.7303462028503418), 0.26965388655662537), lax_reference.div(lax_reference.sub(lax_reference.add(x2, 1), 1.3496379852294922), 0.6503621339797974)]], dtype=object))), 0.5), dtype=object), numpy.array([[ True, True, True], [ True, True, True]]), numpy.array([[False, False, False], [False, False, False]]))' - assert numpy.array_equal( - symbolic_output, - expected_output - ) + expected_output = r"lax_reference.select(numpy.array(lax_reference.gt(lax_reference.select(numpy.numpy.array([[numpy.logical_and(numpy.logical_or(numpy.logical_and(lax_reference.le(lax_reference.abs(lax_reference.sub(0.07225775718688965, x1)), lax_reference.add(9.99999993922529e-09, lax_reference.mul(9.999999747378752e-06, lax_reference.abs(x1)))), numpy.logical_not(numpy.logical_or(False, lax_reference.eq(lax_reference.abs(x1), inf)))), numpy.logical_and(numpy.logical_and(False, lax_reference.eq(lax_reference.abs(x1), inf)), lax_reference.eq(0.07225775718688965, x1))), numpy.logical_not(numpy.logical_or(False, lax_reference.ne(x1, x1)))), numpy.logical_and(numpy.logical_or(numpy.logical_and(lax_reference.le(lax_reference.abs(lax_reference.sub(0.06643760204315186, x1)), lax_reference.add(9.99999993922529e-09, lax_reference.mul(9.999999747378752e-06, lax_reference.abs(x1)))), numpy.logical_not(numpy.logical_or(False, lax_reference.eq(lax_reference.abs(x1), inf)))), numpy.logical_and(numpy.logical_and(False, lax_reference.eq(lax_reference.abs(x1), inf)), lax_reference.eq(0.06643760204315186, x1))), numpy.logical_not(numpy.logical_or(False, lax_reference.ne(x1, x1)))), numpy.logical_and(numpy.logical_or(numpy.logical_and(lax_reference.le(lax_reference.abs(lax_reference.sub(0.9510347843170166, x1)), lax_reference.add(9.99999993922529e-09, lax_reference.mul(9.999999747378752e-06, lax_reference.abs(x1)))), numpy.logical_not(numpy.logical_or(False, lax_reference.eq(lax_reference.abs(x1), inf)))), numpy.logical_and(numpy.logical_and(False, lax_reference.eq(lax_reference.abs(x1), inf)), lax_reference.eq(0.9510347843170166, x1))), numpy.logical_not(numpy.logical_or(False, lax_reference.ne(x1, x1))))], [numpy.logical_and(numpy.logical_or(numpy.logical_and(lax_reference.le(lax_reference.abs(lax_reference.sub(0.8350926637649536, x2)), lax_reference.add(9.99999993922529e-09, lax_reference.mul(9.999999747378752e-06, lax_reference.abs(x2)))), numpy.logical_not(numpy.logical_or(False, lax_reference.eq(lax_reference.abs(x2), inf)))), numpy.logical_and(numpy.logical_and(False, lax_reference.eq(lax_reference.abs(x2), inf)), lax_reference.eq(0.8350926637649536, x2))), numpy.logical_not(numpy.logical_or(False, lax_reference.ne(x2, x2)))), numpy.logical_and(numpy.logical_or(numpy.logical_and(lax_reference.le(lax_reference.abs(lax_reference.sub(0.8651731014251709, x2)), lax_reference.add(9.99999993922529e-09, lax_reference.mul(9.999999747378752e-06, lax_reference.abs(x2)))), numpy.logical_not(numpy.logical_or(False, lax_reference.eq(lax_reference.abs(x2), inf)))), numpy.logical_and(numpy.logical_and(False, lax_reference.eq(lax_reference.abs(x2), inf)), lax_reference.eq(0.8651731014251709, x2))), numpy.logical_not(numpy.logical_or(False, lax_reference.ne(x2, x2)))), numpy.logical_and(numpy.logical_or(numpy.logical_and(lax_reference.le(lax_reference.abs(lax_reference.sub(0.6748189926147461, x2)), lax_reference.add(9.99999993922529e-09, lax_reference.mul(9.999999747378752e-06, lax_reference.abs(x2)))), numpy.logical_not(numpy.logical_or(False, lax_reference.eq(lax_reference.abs(x2), inf)))), numpy.logical_and(numpy.logical_and(False, lax_reference.eq(lax_reference.abs(x2), inf)), lax_reference.eq(0.6748189926147461, x2))), numpy.logical_not(numpy.logical_or(False, lax_reference.ne(x2, x2))))]], dtype=object), numpy.numpy.array([[0.5, 0.5, 0.5], [0.5, 0.5, 0.5]], dtype=numpy.numpy.float32), lax_reference.select(numpy.numpy.array([[lax_reference.lt(x1, 0.07225775718688965), lax_reference.lt(x1, 0.06643760204315186), lax_reference.lt(x1, 0.9510347843170166)], [lax_reference.lt(x2, 0.8350926637649536), lax_reference.lt(x2, 0.8651731014251709), lax_reference.lt(x2, 0.6748189926147461)]], dtype=object), numpy.numpy.array([[lax_reference.div(x1, 0.14451561868190765), lax_reference.div(x1, 0.13287530839443207), lax_reference.div(x1, 1.9020696878433228)], [lax_reference.div(x2, 1.6701854467391968), lax_reference.div(x2, 1.7303463220596313), lax_reference.div(x2, 1.3496381044387817)]], dtype=object), numpy.numpy.array([[lax_reference.div(lax_reference.sub(lax_reference.add(x1, 1), 0.1445155143737793), 1.8554846048355103), lax_reference.div(lax_reference.sub(lax_reference.add(x1, 1), 0.1328752040863037), 1.8671249151229858), lax_reference.div(lax_reference.sub(lax_reference.add(x1, 1), 1.9020695686340332), 0.09793052822351456)], [lax_reference.div(lax_reference.sub(lax_reference.add(x2, 1), 1.6701853275299072), 0.32981476187705994), lax_reference.div(lax_reference.sub(lax_reference.add(x2, 1), 1.7303462028503418), 0.26965388655662537), lax_reference.div(lax_reference.sub(lax_reference.add(x2, 1), 1.3496379852294922), 0.6503621339797974)]], dtype=object))), 0.5), dtype=object), numpy.array([[ True, True, True], [ True, True, True]]), numpy.array([[False, False, False], [False, False, False]]))" + assert numpy.array_equal(symbolic_output, expected_output) # Compute symbolic result with symbolic inputs and symbolic weights symbolic_output = symbolic.apply(symbolic_weights, symbolic_input) @@ -273,8 +274,5 @@ def test_net(type, x): # assert symbolic_output.shape == (2, 3) # Check the form of the symbolic expression # N.B. expected output can change depending due to presence of small numerical errors that can differ between runs and platforms - expected_output = r'lax_reference.select(lax_reference.gt(lax_reference.select(numpy.array([[lax_reference.eq(lax_reference.min(1, lax_reference.max(0, 0.07225776)), x1), lax_reference.eq(lax_reference.min(1, lax_reference.max(0, 0.0664376)), x1), lax_reference.eq(lax_reference.min(1, lax_reference.max(0, 0.9510348)), x1)], [lax_reference.eq(lax_reference.min(1, lax_reference.max(0, 0.83509266)), x2), lax_reference.eq(lax_reference.min(1, lax_reference.max(0, 0.8651731)), x2), lax_reference.eq(lax_reference.min(1, lax_reference.max(0, 0.674819)), x2)]], dtype=object), numpy.array([[0.5, 0.5, 0.5], [0.5, 0.5, 0.5]]), lax_reference.select(numpy.array([[lax_reference.lt(x1, lax_reference.min(1, lax_reference.max(0, 0.07225776))), lax_reference.lt(x1, lax_reference.min(1, lax_reference.max(0, 0.0664376))), lax_reference.lt(x1, lax_reference.min(1, lax_reference.max(0, 0.9510348)))], [lax_reference.lt(x2, lax_reference.min(1, lax_reference.max(0, 0.83509266))), lax_reference.lt(x2, lax_reference.min(1, lax_reference.max(0, 0.8651731))), lax_reference.lt(x2, lax_reference.min(1, lax_reference.max(0, 0.674819)))]], dtype=object), numpy.array([[lax_reference.div(x1, lax_reference.add(lax_reference.mul(2, lax_reference.min(1, lax_reference.max(0, 0.07225776))), 1e-07)), lax_reference.div(x1, lax_reference.add(lax_reference.mul(2, lax_reference.min(1, lax_reference.max(0, 0.0664376))), 1e-07)), lax_reference.div(x1, lax_reference.add(lax_reference.mul(2, lax_reference.min(1, lax_reference.max(0, 0.9510348))), 1e-07))], [lax_reference.div(x2, lax_reference.add(lax_reference.mul(2, lax_reference.min(1, lax_reference.max(0, 0.83509266))), 1e-07)), lax_reference.div(x2, lax_reference.add(lax_reference.mul(2, lax_reference.min(1, lax_reference.max(0, 0.8651731))), 1e-07)), lax_reference.div(x2, lax_reference.add(lax_reference.mul(2, lax_reference.min(1, lax_reference.max(0, 0.674819))), 1e-07))]], dtype=object), numpy.array([[lax_reference.div(lax_reference.sub(lax_reference.add(x1, 1), lax_reference.mul(2, lax_reference.min(1, lax_reference.max(0, 0.07225776)))), lax_reference.add(lax_reference.mul(2, lax_reference.sub(1, lax_reference.min(1, lax_reference.max(0, 0.07225776)))), 1e-07)), lax_reference.div(lax_reference.sub(lax_reference.add(x1, 1), lax_reference.mul(2, lax_reference.min(1, lax_reference.max(0, 0.0664376)))), lax_reference.add(lax_reference.mul(2, lax_reference.sub(1, lax_reference.min(1, lax_reference.max(0, 0.0664376)))), 1e-07)), lax_reference.div(lax_reference.sub(lax_reference.add(x1, 1), lax_reference.mul(2, lax_reference.min(1, lax_reference.max(0, 0.9510348)))), lax_reference.add(lax_reference.mul(2, lax_reference.sub(1, lax_reference.min(1, lax_reference.max(0, 0.9510348)))), 1e-07))], [lax_reference.div(lax_reference.sub(lax_reference.add(x2, 1), lax_reference.mul(2, lax_reference.min(1, lax_reference.max(0, 0.83509266)))), lax_reference.add(lax_reference.mul(2, lax_reference.sub(1, lax_reference.min(1, lax_reference.max(0, 0.83509266)))), 1e-07)), lax_reference.div(lax_reference.sub(lax_reference.add(x2, 1), lax_reference.mul(2, lax_reference.min(1, lax_reference.max(0, 0.8651731)))), lax_reference.add(lax_reference.mul(2, lax_reference.sub(1, lax_reference.min(1, lax_reference.max(0, 0.8651731)))), 1e-07)), lax_reference.div(lax_reference.sub(lax_reference.add(x2, 1), lax_reference.mul(2, lax_reference.min(1, lax_reference.max(0, 0.674819)))), lax_reference.add(lax_reference.mul(2, lax_reference.sub(1, lax_reference.min(1, lax_reference.max(0, 0.674819)))), 1e-07))]], dtype=object))), 0.5), numpy.array([[ True, True, True], [ True, True, True]]), numpy.array([[False, False, False], [False, False, False]]))' - assert numpy.array_equal( - symbolic_output, - expected_output - ) + expected_output = r"lax_reference.select(lax_reference.gt(lax_reference.select(numpy.array([[lax_reference.eq(lax_reference.min(1, lax_reference.max(0, 0.07225776)), x1), lax_reference.eq(lax_reference.min(1, lax_reference.max(0, 0.0664376)), x1), lax_reference.eq(lax_reference.min(1, lax_reference.max(0, 0.9510348)), x1)], [lax_reference.eq(lax_reference.min(1, lax_reference.max(0, 0.83509266)), x2), lax_reference.eq(lax_reference.min(1, lax_reference.max(0, 0.8651731)), x2), lax_reference.eq(lax_reference.min(1, lax_reference.max(0, 0.674819)), x2)]], dtype=object), numpy.array([[0.5, 0.5, 0.5], [0.5, 0.5, 0.5]]), lax_reference.select(numpy.array([[lax_reference.lt(x1, lax_reference.min(1, lax_reference.max(0, 0.07225776))), lax_reference.lt(x1, lax_reference.min(1, lax_reference.max(0, 0.0664376))), lax_reference.lt(x1, lax_reference.min(1, lax_reference.max(0, 0.9510348)))], [lax_reference.lt(x2, lax_reference.min(1, lax_reference.max(0, 0.83509266))), lax_reference.lt(x2, lax_reference.min(1, lax_reference.max(0, 0.8651731))), lax_reference.lt(x2, lax_reference.min(1, lax_reference.max(0, 0.674819)))]], dtype=object), numpy.array([[lax_reference.div(x1, lax_reference.add(lax_reference.mul(2, lax_reference.min(1, lax_reference.max(0, 0.07225776))), 1e-07)), lax_reference.div(x1, lax_reference.add(lax_reference.mul(2, lax_reference.min(1, lax_reference.max(0, 0.0664376))), 1e-07)), lax_reference.div(x1, lax_reference.add(lax_reference.mul(2, lax_reference.min(1, lax_reference.max(0, 0.9510348))), 1e-07))], [lax_reference.div(x2, lax_reference.add(lax_reference.mul(2, lax_reference.min(1, lax_reference.max(0, 0.83509266))), 1e-07)), lax_reference.div(x2, lax_reference.add(lax_reference.mul(2, lax_reference.min(1, lax_reference.max(0, 0.8651731))), 1e-07)), lax_reference.div(x2, lax_reference.add(lax_reference.mul(2, lax_reference.min(1, lax_reference.max(0, 0.674819))), 1e-07))]], dtype=object), numpy.array([[lax_reference.div(lax_reference.sub(lax_reference.add(x1, 1), lax_reference.mul(2, lax_reference.min(1, lax_reference.max(0, 0.07225776)))), lax_reference.add(lax_reference.mul(2, lax_reference.sub(1, lax_reference.min(1, lax_reference.max(0, 0.07225776)))), 1e-07)), lax_reference.div(lax_reference.sub(lax_reference.add(x1, 1), lax_reference.mul(2, lax_reference.min(1, lax_reference.max(0, 0.0664376)))), lax_reference.add(lax_reference.mul(2, lax_reference.sub(1, lax_reference.min(1, lax_reference.max(0, 0.0664376)))), 1e-07)), lax_reference.div(lax_reference.sub(lax_reference.add(x1, 1), lax_reference.mul(2, lax_reference.min(1, lax_reference.max(0, 0.9510348)))), lax_reference.add(lax_reference.mul(2, lax_reference.sub(1, lax_reference.min(1, lax_reference.max(0, 0.9510348)))), 1e-07))], [lax_reference.div(lax_reference.sub(lax_reference.add(x2, 1), lax_reference.mul(2, lax_reference.min(1, lax_reference.max(0, 0.83509266)))), lax_reference.add(lax_reference.mul(2, lax_reference.sub(1, lax_reference.min(1, lax_reference.max(0, 0.83509266)))), 1e-07)), lax_reference.div(lax_reference.sub(lax_reference.add(x2, 1), lax_reference.mul(2, lax_reference.min(1, lax_reference.max(0, 0.8651731)))), lax_reference.add(lax_reference.mul(2, lax_reference.sub(1, lax_reference.min(1, lax_reference.max(0, 0.8651731)))), 1e-07)), lax_reference.div(lax_reference.sub(lax_reference.add(x2, 1), lax_reference.mul(2, lax_reference.min(1, lax_reference.max(0, 0.674819)))), lax_reference.add(lax_reference.mul(2, lax_reference.sub(1, lax_reference.min(1, lax_reference.max(0, 0.674819)))), 1e-07))]], dtype=object))), 0.5), numpy.array([[ True, True, True], [ True, True, True]]), numpy.array([[False, False, False], [False, False, False]]))" + assert numpy.array_equal(symbolic_output, expected_output) From 3a8c64dacc5b02153604736e6fe3a229523d8c29 Mon Sep 17 00:00:00 2001 From: Ian Wright <55286208+Z80coder@users.noreply.github.com> Date: Fri, 24 Feb 2023 15:46:10 +0000 Subject: [PATCH 026/113] make note --- tests/test_noisy_xor.py | 1 + 1 file changed, 1 insertion(+) diff --git a/tests/test_noisy_xor.py b/tests/test_noisy_xor.py index 0260246..097b853 100644 --- a/tests/test_noisy_xor.py +++ b/tests/test_noisy_xor.py @@ -101,6 +101,7 @@ def get_data(): """ +# N.B. 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2023 10:29:31 +0000 Subject: [PATCH 028/113] Update README.md update status icon --- README.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/README.md b/README.md index d76f20c..7606f15 100644 --- a/README.md +++ b/README.md @@ -1,6 +1,6 @@ # differentiable-boolean-logic-networks -[![Python package](https://github.com/github/neural-logic/actions/workflows/python.yaml/badge.svg)](https://github.com/github/neural-logic/actions/workflows/python.yaml) +[![Python package](https://github.com/Z80coder/discrete-differentiable-networks/actions/workflows/python.yaml/badge.svg)](https://github.com/Z80coder/discrete-differentiable-networks/actions/workflows/python.yaml) [![Open in GitHub Codespaces](https://github.com/codespaces/badge.svg)](https://z80coder-legendary-space-cod-65vqgjqqxjq2xv44.github.dev/) From e99bd4bc5ee1b219b0eb518ce61358a02d367f2a Mon Sep 17 00:00:00 2001 From: Ian Wright <55286208+Z80coder@users.noreply.github.com> Date: Mon, 6 Mar 2023 10:57:32 +0000 Subject: [PATCH 029/113] Update README.md --- README.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/README.md b/README.md index 7606f15..1fa4f0a 100644 --- a/README.md +++ b/README.md @@ -1,4 +1,4 @@ -# differentiable-boolean-logic-networks +# discrete-differentiable-networks [![Python package](https://github.com/Z80coder/discrete-differentiable-networks/actions/workflows/python.yaml/badge.svg)](https://github.com/Z80coder/discrete-differentiable-networks/actions/workflows/python.yaml) From bac332ba184bd574f5f796081a74ab30adfac59b Mon Sep 17 00:00:00 2001 From: Ian Wright <55286208+Z80coder@users.noreply.github.com> Date: Mon, 6 Mar 2023 11:41:54 +0000 Subject: [PATCH 030/113] Update README.md Update video links --- README.md | 43 +++++++++++++++++-------------------------- 1 file changed, 17 insertions(+), 26 deletions(-) diff --git a/README.md b/README.md index 1fa4f0a..e6e1c46 100644 --- a/README.md +++ b/README.md @@ -2,44 +2,35 @@ [![Python package](https://github.com/Z80coder/discrete-differentiable-networks/actions/workflows/python.yaml/badge.svg)](https://github.com/Z80coder/discrete-differentiable-networks/actions/workflows/python.yaml) -[![Open in GitHub Codespaces](https://github.com/codespaces/badge.svg)](https://z80coder-legendary-space-cod-65vqgjqqxjq2xv44.github.dev/) - A Boolean Logic Network library for learning boolean functions on GPUs with gradient descent. The working prototype is implemented in Wolfram. The production library is implemented in Python. Questions? Ask @Z80coder -

- -

- ## Demos Neural logic nets -https://user-images.githubusercontent.com/55286208/205712875-54d4cdfe-5199-4429-a438-3b797e4c4456.mp4 - - -- [Neural logic nets for differentiable QL](https://drive.google.com/file/d/195r9Y08Q61V80f2Hqw62YuHpYsCzJCmZ/view?usp=sharing) (30m) -- [Boolean logic nets and MNIST](https://drive.google.com/file/d/1dWAQfFWcOm1ORqfh62H66nigGy2wQ17G/view?usp=share_link) (18m) +- [Neural logic nets]() (15m) ## Development videos -- [The Soft-NOT operator](https://drive.google.com/file/d/1C9egUO9SWSXba7VEqqUPfECYXeLFf5g0/view?usp=sharing) (10m) -- [The Soft-AND operator](https://drive.google.com/file/d/133U60sUh4qzjrieZyMfEzULsZJF27lov/view?usp=sharing) (10m) -- [The differentiable Hard-AND operator](https://drive.google.com/file/d/1cdfMkO0xg-IUYK3avHqfarLJRXGtWcRf/view?usp=sharing) (17m) -- [The differentiable Hard-OR operator](https://drive.google.com/file/d/1v1WMfOWx4PQbjyoPJo82QNh2DGBhM4uH/view?usp=sharing) (5m) -- [The differentiable Hard-MAJORITY operator](https://drive.google.com/file/d/1qVTAFAVZ3Qlk_mYh2wd83uME89RsBzri/view?usp=sharing) (13m) -- [The hardening layer](https://drive.google.com/file/d/1ZEd34UMyFY52_0U2-j58hKJ5uvJBYREn/view?usp=sharing) (11m) -- [The hardening operation](https://drive.google.com/file/d/1M11ovLCbqAfjplFioKpMmX1hOvpwSHXv/view?usp=sharing) (19m) -- [A classifier architecture](https://drive.google.com/file/d/1sQHyo4OjapEj3a0JLhnSYEsLMBMUZMT8/view?usp=sharing) (20m) -- [Neural logic nets](https://drive.google.com/file/d/1P25OxM7Af8ppUGOUhKd6psGHI0OVXIzw/view?usp=sharing) (15m) -- [Learning XOR (parity)](https://drive.google.com/file/d/1kBxJCkuEzbisWhUGJZ42o-m6xYOZ56pB/view?usp=sharing) (10m) -- [Numerical regression](https://drive.google.com/file/d/1k2wQIjTN0omKuaFYQHrusMRIdDxPlSAf/view?usp=sharing) (23m) -- [If-Then-Else neuron](https://drive.google.com/file/d/1qelfWX6s2XhlHxFwUSV76tAS2tyDK3Q0/view?usp=sharing) (23m) -- [Neural conditions and actions](https://drive.google.com/file/d/1nrn_4TlNCmdC1ZAlN9pKIOF2hEjtykuo/view?usp=sharing) (24m) -- [Neural decision lists](https://drive.google.com/file/d/16F_2kpBaZO-qPQLX38Sar9pJfuunsVyO/view?usp=sharing) (15m) -- [Boolean logic nets and MNIST](https://drive.google.com/file/d/1dWAQfFWcOm1ORqfh62H66nigGy2wQ17G/view?usp=share_link) (18m) +- [The Soft-NOT operator](https://drive.google.com/file/d/1_IECuI0f58o_aIIdaQhRo6qPH517YaMa/view?usp=share_link) (10m) +- [The Soft-AND operator](https://drive.google.com/file/d/1l9Y2cWJYYdYSsgqwfH-Dfo2Nxmiewia-/view?usp=share_link) (10m) +- [The differentiable Hard-AND operator](https://drive.google.com/file/d/1Bg1KjKF8KZaBP6jYFhQ5oARrcZYx2O8S/view?usp=share_link) (17m) +- [The differentiable Hard-OR operator](https://drive.google.com/file/d/1WUmJHToU0hQo0YgHlhJb12qECDKzmE8f/view?usp=share_link) (5m) +- [The differentiable Hard-MAJORITY operator](https://drive.google.com/file/d/18oQWhNvbkJGZ49OcQEqGAxkskGZV0e09/view?usp=share_link) (13m) +- [The hardening layer](https://drive.google.com/file/d/1c5K77n9dftsyciq32T7SBBa0PBhIgEq7/view?usp=share_link) (11m) +- [The hardening operation](https://drive.google.com/file/d/1JWA9P9BbfEHWiDfNKVjaH_ssP6CA19Nf/view?usp=share_link) (19m) +- [A classifier architecture](https://drive.google.com/file/d/1KZp8-7hbc_5tHESgmcyBDdBbZDu9UEO9/view?usp=share_link) (20m) +- [Neural logic nets]() (15m) +- [Learning XOR (parity)](https://drive.google.com/file/d/1I2H3iQjM7tNrG83DJFFngQZB_T8jM6uw/view?usp=share_link) (10m) +- [Numerical regression](https://drive.google.com/file/d/1Qx9hBR2nZVymJr3Yoi1CGdg9y8VBxn8P/view?usp=share_link) (23m) +- [If-Then-Else neuron](https://drive.google.com/file/d/1siMqbLr9VYCOwBqNUAnQse9IQSGUjlqo/view?usp=share_link) (23m) +- [Neural conditions and actions](https://drive.google.com/file/d/1WH319bwV55858TYQ9G3C4RPxzdTiA0Ru/view?usp=share_link) (24m) +- [Neural decision lists](https://drive.google.com/file/d/1H0tJtiHz3yXZ7E2xeauaNRd4rnBTUf2v/view?usp=share_link) (15m) +- [Boolean logic nets and MNIST](https://drive.google.com/file/d/12Rwx8H76UTNRdBK4WAwe_QeTWiGrbP-_/view?usp=share_link) (18m) +- [Neural logic nets for differentiable QL](https://drive.google.com/file/d/15rAagCh7LxEN0CHVNkTY6iPWSxrAG0pW/view?usp=share_link) (30m) More to come! From caeb1c3da215c2d9ea2f4bc67a0945331a4d99d1 Mon Sep 17 00:00:00 2001 From: Ian Wright <55286208+Z80coder@users.noreply.github.com> Date: Mon, 6 Mar 2023 11:43:40 +0000 Subject: [PATCH 031/113] Update README.md --- README.md | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/README.md b/README.md index e6e1c46..a60189e 100644 --- a/README.md +++ b/README.md @@ -12,11 +12,11 @@ Questions? Ask @Z80coder Neural logic nets -- [Neural logic nets]() (15m) +- [Neural logic nets](https://drive.google.com/file/d/1_IECuI0f58o_aIIdaQhRo6qPH517YaMa/view?usp=share_link) (15m) ## Development videos -- [The Soft-NOT operator](https://drive.google.com/file/d/1_IECuI0f58o_aIIdaQhRo6qPH517YaMa/view?usp=share_link) (10m) +- [The Soft-NOT operator](https://drive.google.com/file/d/1z2WFpz4eWLb9xauRnIl6mSXhkbU-XR6X/view?usp=share_link) (10m) - [The Soft-AND operator](https://drive.google.com/file/d/1l9Y2cWJYYdYSsgqwfH-Dfo2Nxmiewia-/view?usp=share_link) (10m) - [The differentiable Hard-AND operator](https://drive.google.com/file/d/1Bg1KjKF8KZaBP6jYFhQ5oARrcZYx2O8S/view?usp=share_link) (17m) - [The differentiable Hard-OR operator](https://drive.google.com/file/d/1WUmJHToU0hQo0YgHlhJb12qECDKzmE8f/view?usp=share_link) (5m) @@ -24,7 +24,7 @@ Neural logic nets - [The hardening layer](https://drive.google.com/file/d/1c5K77n9dftsyciq32T7SBBa0PBhIgEq7/view?usp=share_link) (11m) - [The hardening operation](https://drive.google.com/file/d/1JWA9P9BbfEHWiDfNKVjaH_ssP6CA19Nf/view?usp=share_link) (19m) - [A classifier architecture](https://drive.google.com/file/d/1KZp8-7hbc_5tHESgmcyBDdBbZDu9UEO9/view?usp=share_link) (20m) -- [Neural logic nets]() (15m) +- [Neural logic nets](https://drive.google.com/file/d/1_IECuI0f58o_aIIdaQhRo6qPH517YaMa/view?usp=share_link) (15m) - [Learning XOR (parity)](https://drive.google.com/file/d/1I2H3iQjM7tNrG83DJFFngQZB_T8jM6uw/view?usp=share_link) (10m) - [Numerical regression](https://drive.google.com/file/d/1Qx9hBR2nZVymJr3Yoi1CGdg9y8VBxn8P/view?usp=share_link) (23m) - [If-Then-Else neuron](https://drive.google.com/file/d/1siMqbLr9VYCOwBqNUAnQse9IQSGUjlqo/view?usp=share_link) (23m) From 2bdb02f07826075e6222bbe6fa1f0d26d8fbef20 Mon Sep 17 00:00:00 2001 From: Ian Wright Date: Mon, 6 Mar 2023 15:44:57 +0000 Subject: [PATCH 032/113] embryo paper --- docs/db.tex | 200 ++++++++++++++ docs/iclr2021_conference.sty | 246 +++++++++++++++++ docs/math_commands.tex | 508 +++++++++++++++++++++++++++++++++++ 3 files changed, 954 insertions(+) create mode 100644 docs/db.tex create mode 100644 docs/iclr2021_conference.sty create mode 100644 docs/math_commands.tex diff --git a/docs/db.tex b/docs/db.tex new file mode 100644 index 0000000..bc97fd3 --- /dev/null +++ b/docs/db.tex @@ -0,0 +1,200 @@ + +\documentclass{article} % For LaTeX2e +\usepackage{iclr2021_conference,times} + +% Optional math commands from https://github.com/goodfeli/dlbook_notation. +\input{math_commands.tex} + +\usepackage{hyperref} +\usepackage{url} + +\title{Learning boolean functions with $\partial\mathbb{B}$ nets} + +% Authors must not appear in the submitted version. They should be hidden +% as long as the \iclrfinalcopy macro remains commented out below. +% Non-anonymous submissions will be rejected without review. + +\author{Ian Wright \thanks{GitHub, z80coder@github.com} +} + +% The \author macro works with any number of authors. There are two commands +% used to separate the names and addresses of multiple authors: \And and \AND. +% +% Using \And between authors leaves it to \LaTeX{} to determine where to break +% the lines. Using \AND forces a linebreak at that point. So, if \LaTeX{} +% puts 3 of 4 authors names on the first line, and the last on the second +% line, try using \AND instead of \And before the third author name. + +\newcommand{\fix}{\marginpar{FIX}} +\newcommand{\new}{\marginpar{NEW}} + +\iclrfinalcopy % Uncomment for camera-ready version, but NOT for submission. +\begin{document} + + +\maketitle + +\begin{abstract} + $\partial\mathbb{B}$ nets are real-valued, differentiable neural networks that can be `hardened' to generate boolean functions with identical semantics. Existing approaches to neural network binarization lose accuracy; in contrast, training a $\partial\mathbb{B}$ net with backpropagation yields a compact and interpretable boolean function with identical accuracy. Experiments demonstrate that $\partial\mathbb{B}$ nets achieve competitive performance on standard machine learning problems. +\end{abstract} + +\section{Introduction} + +Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum. + + +\begin{figure}[h] +\begin{center} +%\framebox[4.0in]{$\;$} +\fbox{\rule[-.5cm]{0cm}{4cm} \rule[-.5cm]{4cm}{0cm}} +\end{center} +\caption{Sample figure caption.} +\end{figure} + +\section{Tables} + +\begin{table}[h] +\caption{Sample table title} +\label{sample-table} +\begin{center} +\begin{tabular}{ll} +\multicolumn{1}{c}{\bf PART} &\multicolumn{1}{c}{\bf DESCRIPTION} +\\ \hline \\ +Dendrite &Input terminal \\ +Axon &Output terminal \\ +Soma &Cell body (contains cell nucleus) \\ +\end{tabular} +\end{center} +\end{table} + +\section{Default Notation} + + +\centerline{\bf Numbers and Arrays} +\bgroup +\def\arraystretch{1.5} +\begin{tabular}{p{1in}p{3.25in}} +$\displaystyle a$ & A scalar (integer or real)\\ +$\displaystyle \va$ & A vector\\ +$\displaystyle \mA$ & A matrix\\ +$\displaystyle \tA$ & A tensor\\ +$\displaystyle \mI_n$ & Identity matrix with $n$ rows and $n$ columns\\ +$\displaystyle \mI$ & Identity matrix with dimensionality implied by context\\ +$\displaystyle \ve^{(i)}$ & Standard basis vector $[0,\dots,0,1,0,\dots,0]$ with a 1 at position $i$\\ +$\displaystyle \text{diag}(\va)$ & A square, diagonal matrix with diagonal entries given by $\va$\\ +$\displaystyle \ra$ & A scalar random variable\\ +$\displaystyle \rva$ & A vector-valued random variable\\ +$\displaystyle \rmA$ & A matrix-valued random variable\\ +\end{tabular} +\egroup +\vspace{0.25cm} + +\centerline{\bf Sets and Graphs} +\bgroup +\def\arraystretch{1.5} + +\begin{tabular}{p{1.25in}p{3.25in}} +$\displaystyle \sA$ & A set\\ +$\displaystyle \R$ & The set of real numbers \\ +$\displaystyle \{0, 1\}$ & The set containing 0 and 1 \\ +$\displaystyle \{0, 1, \dots, n \}$ & The set of all integers between $0$ and $n$\\ +$\displaystyle [a, b]$ & The real interval including $a$ and $b$\\ +$\displaystyle (a, b]$ & The real interval excluding $a$ but including $b$\\ +$\displaystyle \sA \backslash \sB$ & Set subtraction, i.e., the set containing the elements of $\sA$ that are not in $\sB$\\ +$\displaystyle \gG$ & A graph\\ +$\displaystyle \parents_\gG(\ervx_i)$ & The parents of $\ervx_i$ in $\gG$ +\end{tabular} +\vspace{0.25cm} + + +\centerline{\bf Indexing} +\bgroup +\def\arraystretch{1.5} + +\begin{tabular}{p{1.25in}p{3.25in}} +$\displaystyle \eva_i$ & Element $i$ of vector $\va$, with indexing starting at 1 \\ +$\displaystyle \eva_{-i}$ & All elements of vector $\va$ except for element $i$ \\ +$\displaystyle \emA_{i,j}$ & Element $i, j$ of matrix $\mA$ \\ +$\displaystyle \mA_{i, :}$ & Row $i$ of matrix $\mA$ \\ +$\displaystyle \mA_{:, i}$ & Column $i$ of matrix $\mA$ \\ +$\displaystyle \etA_{i, j, k}$ & Element $(i, j, k)$ of a 3-D tensor $\tA$\\ +$\displaystyle \tA_{:, :, i}$ & 2-D slice of a 3-D tensor\\ +$\displaystyle \erva_i$ & Element $i$ of the random vector $\rva$ \\ +\end{tabular} +\egroup +\vspace{0.25cm} + + +\centerline{\bf Calculus} +\bgroup +\def\arraystretch{1.5} +\begin{tabular}{p{1.25in}p{3.25in}} +% NOTE: the [2ex] on the next line adds extra height to that row of the table. +% Without that command, the fraction on the first line is too tall and collides +% with the fraction on the second line. +$\displaystyle\frac{d y} {d x}$ & Derivative of $y$ with respect to $x$\\ [2ex] +$\displaystyle \frac{\partial y} {\partial x} $ & Partial derivative of $y$ with respect to $x$ \\ +$\displaystyle \nabla_\vx y $ & Gradient of $y$ with respect to $\vx$ \\ +$\displaystyle \nabla_\mX y $ & Matrix derivatives of $y$ with respect to $\mX$ \\ +$\displaystyle \nabla_\tX y $ & Tensor containing derivatives of $y$ with respect to $\tX$ \\ +$\displaystyle \frac{\partial f}{\partial \vx} $ & Jacobian matrix $\mJ \in \R^{m\times n}$ of $f: \R^n \rightarrow \R^m$\\ +$\displaystyle \nabla_\vx^2 f(\vx)\text{ or }\mH( f)(\vx)$ & The Hessian matrix of $f$ at input point $\vx$\\ +$\displaystyle \int f(\vx) d\vx $ & Definite integral over the entire domain of $\vx$ \\ +$\displaystyle \int_\sS f(\vx) d\vx$ & Definite integral with respect to $\vx$ over the set $\sS$ \\ +\end{tabular} +\egroup +\vspace{0.25cm} + +\centerline{\bf Probability and Information Theory} +\bgroup +\def\arraystretch{1.5} +\begin{tabular}{p{1.25in}p{3.25in}} +$\displaystyle P(\ra)$ & A probability distribution over a discrete variable\\ +$\displaystyle p(\ra)$ & A probability distribution over a continuous variable, or over +a variable whose type has not been specified\\ +$\displaystyle \ra \sim P$ & Random variable $\ra$ has distribution $P$\\% so thing on left of \sim should always be a random variable, with name beginning with \r +$\displaystyle \E_{\rx\sim P} [ f(x) ]\text{ or } \E f(x)$ & Expectation of $f(x)$ with respect to $P(\rx)$ \\ +$\displaystyle \Var(f(x)) $ & Variance of $f(x)$ under $P(\rx)$ \\ +$\displaystyle \Cov(f(x),g(x)) $ & Covariance of $f(x)$ and $g(x)$ under $P(\rx)$\\ +$\displaystyle H(\rx) $ & Shannon entropy of the random variable $\rx$\\ +$\displaystyle \KL ( P \Vert Q ) $ & Kullback-Leibler divergence of P and Q \\ +$\displaystyle \mathcal{N} ( \vx ; \vmu , \mSigma)$ & Gaussian distribution % +over $\vx$ with mean $\vmu$ and covariance $\mSigma$ \\ +\end{tabular} +\egroup +\vspace{0.25cm} + +\centerline{\bf Functions} +\bgroup +\def\arraystretch{1.5} +\begin{tabular}{p{1.25in}p{3.25in}} +$\displaystyle f: \sA \rightarrow \sB$ & The function $f$ with domain $\sA$ and range $\sB$\\ +$\displaystyle f \circ g $ & Composition of the functions $f$ and $g$ \\ + $\displaystyle f(\vx ; \vtheta) $ & A function of $\vx$ parametrized by $\vtheta$. + (Sometimes we write $f(\vx)$ and omit the argument $\vtheta$ to lighten notation) \\ +$\displaystyle \log x$ & Natural logarithm of $x$ \\ +$\displaystyle \sigma(x)$ & Logistic sigmoid, $\displaystyle \frac{1} {1 + \exp(-x)}$ \\ +$\displaystyle \zeta(x)$ & Softplus, $\log(1 + \exp(x))$ \\ +$\displaystyle || \vx ||_p $ & $\normlp$ norm of $\vx$ \\ +$\displaystyle || \vx || $ & $\normltwo$ norm of $\vx$ \\ +$\displaystyle x^+$ & Positive part of $x$, i.e., $\max(0,x)$\\ +$\displaystyle \1_\mathrm{condition}$ & is 1 if the condition is true, 0 otherwise\\ +\end{tabular} +\egroup +\vspace{0.25cm} + + + +\subsubsection*{Acknowledgments} +Use unnumbered third level headings for the acknowledgments. All +acknowledgments, including those to funding agencies, go at the end of the paper. + + +\bibliography{iclr2021_conference} +\bibliographystyle{iclr2021_conference} + +\appendix +\section{Appendix} +You may include other additional sections here. + +\end{document} diff --git a/docs/iclr2021_conference.sty b/docs/iclr2021_conference.sty new file mode 100644 index 0000000..752784e --- /dev/null +++ b/docs/iclr2021_conference.sty @@ -0,0 +1,246 @@ +%%%% ICLR Macros (LaTex) +%%%% Adapted by Hugo Larochelle from the NIPS stylefile Macros +%%%% Style File +%%%% Dec 12, 1990 Rev Aug 14, 1991; Sept, 1995; April, 1997; April, 1999; October 2014 + +% This file can be used with Latex2e whether running in main mode, or +% 2.09 compatibility mode. +% +% If using main mode, you need to include the commands +% \documentclass{article} +% \usepackage{iclr14submit_e,times} +% + +% Change the overall width of the page. If these parameters are +% changed, they will require corresponding changes in the +% maketitle section. +% +\usepackage{eso-pic} % used by \AddToShipoutPicture +\RequirePackage{fancyhdr} +\RequirePackage{natbib} + +% modification to natbib citations +\setcitestyle{authoryear,round,citesep={;},aysep={,},yysep={;}} + +\renewcommand{\topfraction}{0.95} % let figure take up nearly whole page +\renewcommand{\textfraction}{0.05} % let figure take up nearly whole page + +% Define iclrfinal, set to true if iclrfinalcopy is defined +\newif\ificlrfinal +\iclrfinalfalse +\def\iclrfinalcopy{\iclrfinaltrue} +\font\iclrtenhv = phvb at 8pt + +% Specify the dimensions of each page + +\setlength{\paperheight}{11in} +\setlength{\paperwidth}{8.5in} + + +\oddsidemargin .5in % Note \oddsidemargin = \evensidemargin +\evensidemargin .5in +\marginparwidth 0.07 true in +%\marginparwidth 0.75 true in +%\topmargin 0 true pt % Nominal distance from top of page to top of +%\topmargin 0.125in +\topmargin -0.625in +\addtolength{\headsep}{0.25in} +\textheight 9.0 true in % Height of text (including footnotes & figures) +\textwidth 5.5 true in % Width of text line. +\widowpenalty=10000 +\clubpenalty=10000 + +% \thispagestyle{empty} \pagestyle{empty} +\flushbottom \sloppy + +% We're never going to need a table of contents, so just flush it to +% save space --- suggested by drstrip@sandia-2 +\def\addcontentsline#1#2#3{} + +% Title stuff, taken from deproc. +\def\maketitle{\par +\begingroup + \def\thefootnote{\fnsymbol{footnote}} + \def\@makefnmark{\hbox to 0pt{$^{\@thefnmark}$\hss}} % for perfect author + % name centering +% The footnote-mark was overlapping the footnote-text, +% added the following to fix this problem (MK) + \long\def\@makefntext##1{\parindent 1em\noindent + \hbox to1.8em{\hss $\m@th ^{\@thefnmark}$}##1} + \@maketitle \@thanks +\endgroup +\setcounter{footnote}{0} +\let\maketitle\relax \let\@maketitle\relax +\gdef\@thanks{}\gdef\@author{}\gdef\@title{}\let\thanks\relax} + +% The toptitlebar has been raised to top-justify the first page + +\usepackage{fancyhdr} +\pagestyle{fancy} +\fancyhead{} + +% Title (includes both anonimized and non-anonimized versions) +\def\@maketitle{\vbox{\hsize\textwidth +%\linewidth\hsize \vskip 0.1in \toptitlebar \centering +{\LARGE\sc \@title\par} +%\bottomtitlebar % \vskip 0.1in % minus +\ificlrfinal +% \lhead{Published as a conference paper at ICLR 2021} + \lhead{draft} + \def\And{\end{tabular}\hfil\linebreak[0]\hfil + \begin{tabular}[t]{l}\bf\rule{\z@}{24pt}\ignorespaces}% + \def\AND{\end{tabular}\hfil\linebreak[4]\hfil + \begin{tabular}[t]{l}\bf\rule{\z@}{24pt}\ignorespaces}% + \begin{tabular}[t]{l}\bf\rule{\z@}{24pt}\@author\end{tabular}% +\else + \lhead{draft} + \def\And{\end{tabular}\hfil\linebreak[0]\hfil + \begin{tabular}[t]{l}\bf\rule{\z@}{24pt}\ignorespaces}% + \def\AND{\end{tabular}\hfil\linebreak[4]\hfil + \begin{tabular}[t]{l}\bf\rule{\z@}{24pt}\ignorespaces}% + \begin{tabular}[t]{l}\bf\rule{\z@}{24pt}Anonymous authors\\Paper under double-blind review\end{tabular}% +\fi +\vskip 0.3in minus 0.1in}} + +\renewenvironment{abstract}{\vskip.075in\centerline{\large\sc +Abstract}\vspace{0.5ex}\begin{quote}}{\par\end{quote}\vskip 1ex} + +% sections with less space +\def\section{\@startsection {section}{1}{\z@}{-2.0ex plus + -0.5ex minus -.2ex}{1.5ex plus 0.3ex +minus0.2ex}{\large\sc\raggedright}} + +\def\subsection{\@startsection{subsection}{2}{\z@}{-1.8ex plus +-0.5ex minus -.2ex}{0.8ex plus .2ex}{\normalsize\sc\raggedright}} +\def\subsubsection{\@startsection{subsubsection}{3}{\z@}{-1.5ex +plus -0.5ex minus -.2ex}{0.5ex plus +.2ex}{\normalsize\sc\raggedright}} +\def\paragraph{\@startsection{paragraph}{4}{\z@}{1.5ex plus +0.5ex minus .2ex}{-1em}{\normalsize\bf}} +\def\subparagraph{\@startsection{subparagraph}{5}{\z@}{1.5ex plus + 0.5ex minus .2ex}{-1em}{\normalsize\sc}} +\def\subsubsubsection{\vskip +5pt{\noindent\normalsize\rm\raggedright}} + + +% Footnotes +\footnotesep 6.65pt % +\skip\footins 9pt plus 4pt minus 2pt +\def\footnoterule{\kern-3pt \hrule width 12pc \kern 2.6pt } +\setcounter{footnote}{0} + +% Lists and paragraphs +\parindent 0pt +\topsep 4pt plus 1pt minus 2pt +\partopsep 1pt plus 0.5pt minus 0.5pt +\itemsep 2pt plus 1pt minus 0.5pt +\parsep 2pt plus 1pt minus 0.5pt +\parskip .5pc + + +%\leftmargin2em +\leftmargin3pc +\leftmargini\leftmargin \leftmarginii 2em +\leftmarginiii 1.5em \leftmarginiv 1.0em \leftmarginv .5em + +%\labelsep \labelsep 5pt + +\def\@listi{\leftmargin\leftmargini} +\def\@listii{\leftmargin\leftmarginii + \labelwidth\leftmarginii\advance\labelwidth-\labelsep + \topsep 2pt plus 1pt minus 0.5pt + \parsep 1pt plus 0.5pt minus 0.5pt + \itemsep \parsep} +\def\@listiii{\leftmargin\leftmarginiii + \labelwidth\leftmarginiii\advance\labelwidth-\labelsep + \topsep 1pt plus 0.5pt minus 0.5pt + \parsep \z@ \partopsep 0.5pt plus 0pt minus 0.5pt + \itemsep \topsep} +\def\@listiv{\leftmargin\leftmarginiv + \labelwidth\leftmarginiv\advance\labelwidth-\labelsep} +\def\@listv{\leftmargin\leftmarginv + \labelwidth\leftmarginv\advance\labelwidth-\labelsep} +\def\@listvi{\leftmargin\leftmarginvi + \labelwidth\leftmarginvi\advance\labelwidth-\labelsep} + +\abovedisplayskip 7pt plus2pt minus5pt% +\belowdisplayskip \abovedisplayskip +\abovedisplayshortskip 0pt plus3pt% +\belowdisplayshortskip 4pt plus3pt minus3pt% + +% Less leading in most fonts (due to the narrow columns) +% The choices were between 1-pt and 1.5-pt leading +%\def\@normalsize{\@setsize\normalsize{11pt}\xpt\@xpt} % got rid of @ (MK) +\def\normalsize{\@setsize\normalsize{11pt}\xpt\@xpt} +\def\small{\@setsize\small{10pt}\ixpt\@ixpt} +\def\footnotesize{\@setsize\footnotesize{10pt}\ixpt\@ixpt} +\def\scriptsize{\@setsize\scriptsize{8pt}\viipt\@viipt} +\def\tiny{\@setsize\tiny{7pt}\vipt\@vipt} +\def\large{\@setsize\large{14pt}\xiipt\@xiipt} +\def\Large{\@setsize\Large{16pt}\xivpt\@xivpt} +\def\LARGE{\@setsize\LARGE{20pt}\xviipt\@xviipt} +\def\huge{\@setsize\huge{23pt}\xxpt\@xxpt} +\def\Huge{\@setsize\Huge{28pt}\xxvpt\@xxvpt} + +\def\toptitlebar{\hrule height4pt\vskip .25in\vskip-\parskip} + +\def\bottomtitlebar{\vskip .29in\vskip-\parskip\hrule height1pt\vskip +.09in} % +%Reduced second vskip to compensate for adding the strut in \@author + + +%% % Vertical Ruler +%% % This code is, largely, from the CVPR 2010 conference style file +%% % ----- define vruler +%% \makeatletter +%% \newbox\iclrrulerbox +%% \newcount\iclrrulercount +%% \newdimen\iclrruleroffset +%% \newdimen\cv@lineheight +%% \newdimen\cv@boxheight +%% \newbox\cv@tmpbox +%% \newcount\cv@refno +%% \newcount\cv@tot +%% % NUMBER with left flushed zeros \fillzeros[] +%% \newcount\cv@tmpc@ \newcount\cv@tmpc +%% \def\fillzeros[#1]#2{\cv@tmpc@=#2\relax\ifnum\cv@tmpc@<0\cv@tmpc@=-\cv@tmpc@\fi +%% \cv@tmpc=1 % +%% \loop\ifnum\cv@tmpc@<10 \else \divide\cv@tmpc@ by 10 \advance\cv@tmpc by 1 \fi +%% \ifnum\cv@tmpc@=10\relax\cv@tmpc@=11\relax\fi \ifnum\cv@tmpc@>10 \repeat +%% \ifnum#2<0\advance\cv@tmpc1\relax-\fi +%% \loop\ifnum\cv@tmpc<#1\relax0\advance\cv@tmpc1\relax\fi \ifnum\cv@tmpc<#1 \repeat +%% \cv@tmpc@=#2\relax\ifnum\cv@tmpc@<0\cv@tmpc@=-\cv@tmpc@\fi \relax\the\cv@tmpc@}% +%% % \makevruler[][][][][] +%% \def\makevruler[#1][#2][#3][#4][#5]{\begingroup\offinterlineskip +%% \textheight=#5\vbadness=10000\vfuzz=120ex\overfullrule=0pt% +%% \global\setbox\iclrrulerbox=\vbox to \textheight{% +%% {\parskip=0pt\hfuzz=150em\cv@boxheight=\textheight +%% \cv@lineheight=#1\global\iclrrulercount=#2% +%% \cv@tot\cv@boxheight\divide\cv@tot\cv@lineheight\advance\cv@tot2% +%% \cv@refno1\vskip-\cv@lineheight\vskip1ex% +%% \loop\setbox\cv@tmpbox=\hbox to0cm{{\iclrtenhv\hfil\fillzeros[#4]\iclrrulercount}}% +%% \ht\cv@tmpbox\cv@lineheight\dp\cv@tmpbox0pt\box\cv@tmpbox\break +%% \advance\cv@refno1\global\advance\iclrrulercount#3\relax +%% \ifnum\cv@refno<\cv@tot\repeat}}\endgroup}% +%% \makeatother +%% % ----- end of vruler + +%% % \makevruler[][][][][] +%% \def\iclrruler#1{\makevruler[12pt][#1][1][3][0.993\textheight]\usebox{\iclrrulerbox}} +%% \AddToShipoutPicture{% +%% \ificlrfinal\else +%% \iclrruleroffset=\textheight +%% \advance\iclrruleroffset by -3.7pt +%% \color[rgb]{.7,.7,.7} +%% \AtTextUpperLeft{% +%% \put(\LenToUnit{-35pt},\LenToUnit{-\iclrruleroffset}){%left ruler +%% \iclrruler{\iclrrulercount}} +%% } +%% \fi +%% } +%%% To add a vertical bar on the side +%\AddToShipoutPicture{ +%\AtTextLowerLeft{ +%\hspace*{-1.8cm} +%\colorbox[rgb]{0.7,0.7,0.7}{\small \parbox[b][\textheight]{0.1cm}{}}} +%} diff --git a/docs/math_commands.tex b/docs/math_commands.tex new file mode 100644 index 0000000..0668f93 --- /dev/null +++ b/docs/math_commands.tex @@ -0,0 +1,508 @@ +%%%%% NEW MATH DEFINITIONS %%%%% + +\usepackage{amsmath,amsfonts,bm} + +% Mark sections of captions for referring to divisions of figures +\newcommand{\figleft}{{\em (Left)}} +\newcommand{\figcenter}{{\em (Center)}} +\newcommand{\figright}{{\em (Right)}} +\newcommand{\figtop}{{\em (Top)}} +\newcommand{\figbottom}{{\em (Bottom)}} +\newcommand{\captiona}{{\em (a)}} +\newcommand{\captionb}{{\em (b)}} +\newcommand{\captionc}{{\em (c)}} +\newcommand{\captiond}{{\em (d)}} + +% Highlight a newly defined term +\newcommand{\newterm}[1]{{\bf #1}} + + +% Figure reference, lower-case. +\def\figref#1{figure~\ref{#1}} +% Figure reference, capital. For start of sentence +\def\Figref#1{Figure~\ref{#1}} +\def\twofigref#1#2{figures \ref{#1} and \ref{#2}} +\def\quadfigref#1#2#3#4{figures \ref{#1}, \ref{#2}, \ref{#3} and \ref{#4}} +% Section reference, lower-case. +\def\secref#1{section~\ref{#1}} +% Section reference, capital. +\def\Secref#1{Section~\ref{#1}} +% Reference to two sections. +\def\twosecrefs#1#2{sections \ref{#1} and \ref{#2}} +% Reference to three sections. +\def\secrefs#1#2#3{sections \ref{#1}, \ref{#2} and \ref{#3}} +% Reference to an equation, lower-case. +\def\eqref#1{equation~\ref{#1}} +% Reference to an equation, upper case +\def\Eqref#1{Equation~\ref{#1}} +% A raw reference to an equation---avoid using if possible +\def\plaineqref#1{\ref{#1}} +% Reference to a chapter, lower-case. +\def\chapref#1{chapter~\ref{#1}} +% Reference to an equation, upper case. +\def\Chapref#1{Chapter~\ref{#1}} +% Reference to a range of chapters +\def\rangechapref#1#2{chapters\ref{#1}--\ref{#2}} +% Reference to an algorithm, lower-case. +\def\algref#1{algorithm~\ref{#1}} +% Reference to an algorithm, upper case. +\def\Algref#1{Algorithm~\ref{#1}} +\def\twoalgref#1#2{algorithms \ref{#1} and \ref{#2}} +\def\Twoalgref#1#2{Algorithms \ref{#1} and \ref{#2}} +% Reference to a part, lower case +\def\partref#1{part~\ref{#1}} +% Reference to a part, upper case +\def\Partref#1{Part~\ref{#1}} +\def\twopartref#1#2{parts \ref{#1} and \ref{#2}} + +\def\ceil#1{\lceil #1 \rceil} +\def\floor#1{\lfloor #1 \rfloor} +\def\1{\bm{1}} +\newcommand{\train}{\mathcal{D}} +\newcommand{\valid}{\mathcal{D_{\mathrm{valid}}}} +\newcommand{\test}{\mathcal{D_{\mathrm{test}}}} + +\def\eps{{\epsilon}} + + +% Random variables +\def\reta{{\textnormal{$\eta$}}} +\def\ra{{\textnormal{a}}} +\def\rb{{\textnormal{b}}} +\def\rc{{\textnormal{c}}} +\def\rd{{\textnormal{d}}} +\def\re{{\textnormal{e}}} +\def\rf{{\textnormal{f}}} +\def\rg{{\textnormal{g}}} +\def\rh{{\textnormal{h}}} +\def\ri{{\textnormal{i}}} +\def\rj{{\textnormal{j}}} +\def\rk{{\textnormal{k}}} +\def\rl{{\textnormal{l}}} +% rm is already a command, just don't name any random variables m +\def\rn{{\textnormal{n}}} +\def\ro{{\textnormal{o}}} +\def\rp{{\textnormal{p}}} +\def\rq{{\textnormal{q}}} +\def\rr{{\textnormal{r}}} +\def\rs{{\textnormal{s}}} +\def\rt{{\textnormal{t}}} +\def\ru{{\textnormal{u}}} +\def\rv{{\textnormal{v}}} +\def\rw{{\textnormal{w}}} +\def\rx{{\textnormal{x}}} +\def\ry{{\textnormal{y}}} +\def\rz{{\textnormal{z}}} + +% Random vectors +\def\rvepsilon{{\mathbf{\epsilon}}} +\def\rvtheta{{\mathbf{\theta}}} +\def\rva{{\mathbf{a}}} +\def\rvb{{\mathbf{b}}} +\def\rvc{{\mathbf{c}}} +\def\rvd{{\mathbf{d}}} +\def\rve{{\mathbf{e}}} +\def\rvf{{\mathbf{f}}} +\def\rvg{{\mathbf{g}}} +\def\rvh{{\mathbf{h}}} +\def\rvu{{\mathbf{i}}} +\def\rvj{{\mathbf{j}}} +\def\rvk{{\mathbf{k}}} +\def\rvl{{\mathbf{l}}} +\def\rvm{{\mathbf{m}}} +\def\rvn{{\mathbf{n}}} +\def\rvo{{\mathbf{o}}} +\def\rvp{{\mathbf{p}}} +\def\rvq{{\mathbf{q}}} +\def\rvr{{\mathbf{r}}} +\def\rvs{{\mathbf{s}}} +\def\rvt{{\mathbf{t}}} +\def\rvu{{\mathbf{u}}} +\def\rvv{{\mathbf{v}}} +\def\rvw{{\mathbf{w}}} +\def\rvx{{\mathbf{x}}} +\def\rvy{{\mathbf{y}}} +\def\rvz{{\mathbf{z}}} + +% Elements of random vectors +\def\erva{{\textnormal{a}}} +\def\ervb{{\textnormal{b}}} +\def\ervc{{\textnormal{c}}} +\def\ervd{{\textnormal{d}}} +\def\erve{{\textnormal{e}}} +\def\ervf{{\textnormal{f}}} +\def\ervg{{\textnormal{g}}} +\def\ervh{{\textnormal{h}}} +\def\ervi{{\textnormal{i}}} +\def\ervj{{\textnormal{j}}} +\def\ervk{{\textnormal{k}}} +\def\ervl{{\textnormal{l}}} +\def\ervm{{\textnormal{m}}} +\def\ervn{{\textnormal{n}}} +\def\ervo{{\textnormal{o}}} +\def\ervp{{\textnormal{p}}} +\def\ervq{{\textnormal{q}}} +\def\ervr{{\textnormal{r}}} +\def\ervs{{\textnormal{s}}} +\def\ervt{{\textnormal{t}}} +\def\ervu{{\textnormal{u}}} +\def\ervv{{\textnormal{v}}} +\def\ervw{{\textnormal{w}}} +\def\ervx{{\textnormal{x}}} +\def\ervy{{\textnormal{y}}} +\def\ervz{{\textnormal{z}}} + +% Random matrices +\def\rmA{{\mathbf{A}}} +\def\rmB{{\mathbf{B}}} +\def\rmC{{\mathbf{C}}} +\def\rmD{{\mathbf{D}}} +\def\rmE{{\mathbf{E}}} +\def\rmF{{\mathbf{F}}} +\def\rmG{{\mathbf{G}}} +\def\rmH{{\mathbf{H}}} +\def\rmI{{\mathbf{I}}} +\def\rmJ{{\mathbf{J}}} +\def\rmK{{\mathbf{K}}} +\def\rmL{{\mathbf{L}}} +\def\rmM{{\mathbf{M}}} +\def\rmN{{\mathbf{N}}} +\def\rmO{{\mathbf{O}}} +\def\rmP{{\mathbf{P}}} +\def\rmQ{{\mathbf{Q}}} +\def\rmR{{\mathbf{R}}} +\def\rmS{{\mathbf{S}}} +\def\rmT{{\mathbf{T}}} +\def\rmU{{\mathbf{U}}} +\def\rmV{{\mathbf{V}}} +\def\rmW{{\mathbf{W}}} +\def\rmX{{\mathbf{X}}} +\def\rmY{{\mathbf{Y}}} +\def\rmZ{{\mathbf{Z}}} + +% Elements of random matrices +\def\ermA{{\textnormal{A}}} +\def\ermB{{\textnormal{B}}} +\def\ermC{{\textnormal{C}}} +\def\ermD{{\textnormal{D}}} +\def\ermE{{\textnormal{E}}} +\def\ermF{{\textnormal{F}}} +\def\ermG{{\textnormal{G}}} +\def\ermH{{\textnormal{H}}} +\def\ermI{{\textnormal{I}}} +\def\ermJ{{\textnormal{J}}} +\def\ermK{{\textnormal{K}}} +\def\ermL{{\textnormal{L}}} +\def\ermM{{\textnormal{M}}} +\def\ermN{{\textnormal{N}}} +\def\ermO{{\textnormal{O}}} +\def\ermP{{\textnormal{P}}} +\def\ermQ{{\textnormal{Q}}} +\def\ermR{{\textnormal{R}}} +\def\ermS{{\textnormal{S}}} +\def\ermT{{\textnormal{T}}} +\def\ermU{{\textnormal{U}}} +\def\ermV{{\textnormal{V}}} +\def\ermW{{\textnormal{W}}} +\def\ermX{{\textnormal{X}}} +\def\ermY{{\textnormal{Y}}} +\def\ermZ{{\textnormal{Z}}} + +% Vectors +\def\vzero{{\bm{0}}} +\def\vone{{\bm{1}}} +\def\vmu{{\bm{\mu}}} +\def\vtheta{{\bm{\theta}}} +\def\va{{\bm{a}}} +\def\vb{{\bm{b}}} +\def\vc{{\bm{c}}} +\def\vd{{\bm{d}}} +\def\ve{{\bm{e}}} +\def\vf{{\bm{f}}} +\def\vg{{\bm{g}}} +\def\vh{{\bm{h}}} +\def\vi{{\bm{i}}} +\def\vj{{\bm{j}}} +\def\vk{{\bm{k}}} +\def\vl{{\bm{l}}} +\def\vm{{\bm{m}}} +\def\vn{{\bm{n}}} +\def\vo{{\bm{o}}} +\def\vp{{\bm{p}}} +\def\vq{{\bm{q}}} +\def\vr{{\bm{r}}} +\def\vs{{\bm{s}}} +\def\vt{{\bm{t}}} +\def\vu{{\bm{u}}} +\def\vv{{\bm{v}}} +\def\vw{{\bm{w}}} +\def\vx{{\bm{x}}} +\def\vy{{\bm{y}}} +\def\vz{{\bm{z}}} + +% Elements of vectors +\def\evalpha{{\alpha}} +\def\evbeta{{\beta}} +\def\evepsilon{{\epsilon}} +\def\evlambda{{\lambda}} +\def\evomega{{\omega}} +\def\evmu{{\mu}} +\def\evpsi{{\psi}} +\def\evsigma{{\sigma}} +\def\evtheta{{\theta}} +\def\eva{{a}} +\def\evb{{b}} +\def\evc{{c}} +\def\evd{{d}} +\def\eve{{e}} +\def\evf{{f}} +\def\evg{{g}} +\def\evh{{h}} +\def\evi{{i}} +\def\evj{{j}} +\def\evk{{k}} +\def\evl{{l}} +\def\evm{{m}} +\def\evn{{n}} +\def\evo{{o}} +\def\evp{{p}} +\def\evq{{q}} +\def\evr{{r}} +\def\evs{{s}} +\def\evt{{t}} +\def\evu{{u}} +\def\evv{{v}} +\def\evw{{w}} +\def\evx{{x}} +\def\evy{{y}} +\def\evz{{z}} + +% Matrix +\def\mA{{\bm{A}}} +\def\mB{{\bm{B}}} +\def\mC{{\bm{C}}} +\def\mD{{\bm{D}}} +\def\mE{{\bm{E}}} +\def\mF{{\bm{F}}} +\def\mG{{\bm{G}}} +\def\mH{{\bm{H}}} +\def\mI{{\bm{I}}} +\def\mJ{{\bm{J}}} +\def\mK{{\bm{K}}} +\def\mL{{\bm{L}}} +\def\mM{{\bm{M}}} +\def\mN{{\bm{N}}} +\def\mO{{\bm{O}}} +\def\mP{{\bm{P}}} +\def\mQ{{\bm{Q}}} +\def\mR{{\bm{R}}} +\def\mS{{\bm{S}}} +\def\mT{{\bm{T}}} +\def\mU{{\bm{U}}} +\def\mV{{\bm{V}}} +\def\mW{{\bm{W}}} +\def\mX{{\bm{X}}} +\def\mY{{\bm{Y}}} +\def\mZ{{\bm{Z}}} +\def\mBeta{{\bm{\beta}}} +\def\mPhi{{\bm{\Phi}}} +\def\mLambda{{\bm{\Lambda}}} +\def\mSigma{{\bm{\Sigma}}} + +% Tensor +\DeclareMathAlphabet{\mathsfit}{\encodingdefault}{\sfdefault}{m}{sl} +\SetMathAlphabet{\mathsfit}{bold}{\encodingdefault}{\sfdefault}{bx}{n} +\newcommand{\tens}[1]{\bm{\mathsfit{#1}}} +\def\tA{{\tens{A}}} +\def\tB{{\tens{B}}} +\def\tC{{\tens{C}}} +\def\tD{{\tens{D}}} +\def\tE{{\tens{E}}} +\def\tF{{\tens{F}}} +\def\tG{{\tens{G}}} +\def\tH{{\tens{H}}} +\def\tI{{\tens{I}}} +\def\tJ{{\tens{J}}} +\def\tK{{\tens{K}}} +\def\tL{{\tens{L}}} +\def\tM{{\tens{M}}} +\def\tN{{\tens{N}}} +\def\tO{{\tens{O}}} +\def\tP{{\tens{P}}} +\def\tQ{{\tens{Q}}} +\def\tR{{\tens{R}}} +\def\tS{{\tens{S}}} +\def\tT{{\tens{T}}} +\def\tU{{\tens{U}}} +\def\tV{{\tens{V}}} +\def\tW{{\tens{W}}} +\def\tX{{\tens{X}}} +\def\tY{{\tens{Y}}} +\def\tZ{{\tens{Z}}} + + +% Graph +\def\gA{{\mathcal{A}}} +\def\gB{{\mathcal{B}}} +\def\gC{{\mathcal{C}}} +\def\gD{{\mathcal{D}}} +\def\gE{{\mathcal{E}}} +\def\gF{{\mathcal{F}}} +\def\gG{{\mathcal{G}}} +\def\gH{{\mathcal{H}}} +\def\gI{{\mathcal{I}}} +\def\gJ{{\mathcal{J}}} +\def\gK{{\mathcal{K}}} +\def\gL{{\mathcal{L}}} +\def\gM{{\mathcal{M}}} +\def\gN{{\mathcal{N}}} +\def\gO{{\mathcal{O}}} +\def\gP{{\mathcal{P}}} +\def\gQ{{\mathcal{Q}}} +\def\gR{{\mathcal{R}}} +\def\gS{{\mathcal{S}}} +\def\gT{{\mathcal{T}}} +\def\gU{{\mathcal{U}}} +\def\gV{{\mathcal{V}}} +\def\gW{{\mathcal{W}}} +\def\gX{{\mathcal{X}}} +\def\gY{{\mathcal{Y}}} +\def\gZ{{\mathcal{Z}}} + +% Sets +\def\sA{{\mathbb{A}}} +\def\sB{{\mathbb{B}}} +\def\sC{{\mathbb{C}}} +\def\sD{{\mathbb{D}}} +% Don't use a set called E, because this would be the same as our symbol +% for expectation. +\def\sF{{\mathbb{F}}} +\def\sG{{\mathbb{G}}} +\def\sH{{\mathbb{H}}} +\def\sI{{\mathbb{I}}} +\def\sJ{{\mathbb{J}}} +\def\sK{{\mathbb{K}}} +\def\sL{{\mathbb{L}}} +\def\sM{{\mathbb{M}}} +\def\sN{{\mathbb{N}}} +\def\sO{{\mathbb{O}}} +\def\sP{{\mathbb{P}}} +\def\sQ{{\mathbb{Q}}} +\def\sR{{\mathbb{R}}} +\def\sS{{\mathbb{S}}} +\def\sT{{\mathbb{T}}} +\def\sU{{\mathbb{U}}} +\def\sV{{\mathbb{V}}} +\def\sW{{\mathbb{W}}} +\def\sX{{\mathbb{X}}} +\def\sY{{\mathbb{Y}}} +\def\sZ{{\mathbb{Z}}} + +% Entries of a matrix +\def\emLambda{{\Lambda}} +\def\emA{{A}} +\def\emB{{B}} +\def\emC{{C}} +\def\emD{{D}} +\def\emE{{E}} +\def\emF{{F}} +\def\emG{{G}} +\def\emH{{H}} +\def\emI{{I}} +\def\emJ{{J}} +\def\emK{{K}} +\def\emL{{L}} +\def\emM{{M}} +\def\emN{{N}} +\def\emO{{O}} +\def\emP{{P}} +\def\emQ{{Q}} +\def\emR{{R}} +\def\emS{{S}} +\def\emT{{T}} +\def\emU{{U}} +\def\emV{{V}} +\def\emW{{W}} +\def\emX{{X}} +\def\emY{{Y}} +\def\emZ{{Z}} +\def\emSigma{{\Sigma}} + +% entries of a tensor +% Same font as tensor, without \bm wrapper +\newcommand{\etens}[1]{\mathsfit{#1}} +\def\etLambda{{\etens{\Lambda}}} +\def\etA{{\etens{A}}} +\def\etB{{\etens{B}}} +\def\etC{{\etens{C}}} +\def\etD{{\etens{D}}} +\def\etE{{\etens{E}}} +\def\etF{{\etens{F}}} +\def\etG{{\etens{G}}} +\def\etH{{\etens{H}}} +\def\etI{{\etens{I}}} +\def\etJ{{\etens{J}}} +\def\etK{{\etens{K}}} +\def\etL{{\etens{L}}} +\def\etM{{\etens{M}}} +\def\etN{{\etens{N}}} +\def\etO{{\etens{O}}} +\def\etP{{\etens{P}}} +\def\etQ{{\etens{Q}}} +\def\etR{{\etens{R}}} +\def\etS{{\etens{S}}} +\def\etT{{\etens{T}}} +\def\etU{{\etens{U}}} +\def\etV{{\etens{V}}} +\def\etW{{\etens{W}}} +\def\etX{{\etens{X}}} +\def\etY{{\etens{Y}}} +\def\etZ{{\etens{Z}}} + +% The true underlying data generating distribution +\newcommand{\pdata}{p_{\rm{data}}} +% The empirical distribution defined by the training set +\newcommand{\ptrain}{\hat{p}_{\rm{data}}} +\newcommand{\Ptrain}{\hat{P}_{\rm{data}}} +% The model distribution +\newcommand{\pmodel}{p_{\rm{model}}} +\newcommand{\Pmodel}{P_{\rm{model}}} +\newcommand{\ptildemodel}{\tilde{p}_{\rm{model}}} +% Stochastic autoencoder distributions +\newcommand{\pencode}{p_{\rm{encoder}}} +\newcommand{\pdecode}{p_{\rm{decoder}}} +\newcommand{\precons}{p_{\rm{reconstruct}}} + +\newcommand{\laplace}{\mathrm{Laplace}} % Laplace distribution + +\newcommand{\E}{\mathbb{E}} +\newcommand{\Ls}{\mathcal{L}} +\newcommand{\R}{\mathbb{R}} +\newcommand{\emp}{\tilde{p}} +\newcommand{\lr}{\alpha} +\newcommand{\reg}{\lambda} +\newcommand{\rect}{\mathrm{rectifier}} +\newcommand{\softmax}{\mathrm{softmax}} +\newcommand{\sigmoid}{\sigma} +\newcommand{\softplus}{\zeta} +\newcommand{\KL}{D_{\mathrm{KL}}} +\newcommand{\Var}{\mathrm{Var}} +\newcommand{\standarderror}{\mathrm{SE}} +\newcommand{\Cov}{\mathrm{Cov}} +% Wolfram Mathworld says $L^2$ is for function spaces and $\ell^2$ is for vectors +% But then they seem to use $L^2$ for vectors throughout the site, and so does +% wikipedia. +\newcommand{\normlzero}{L^0} +\newcommand{\normlone}{L^1} +\newcommand{\normltwo}{L^2} +\newcommand{\normlp}{L^p} +\newcommand{\normmax}{L^\infty} + +\newcommand{\parents}{Pa} % See usage in notation.tex. Chosen to match Daphne's book. + +\DeclareMathOperator*{\argmax}{arg\,max} +\DeclareMathOperator*{\argmin}{arg\,min} + +\DeclareMathOperator{\sign}{sign} +\DeclareMathOperator{\Tr}{Tr} +\let\ab\allowbreak From f42868e7aae7ef8ad85cc0d4a0a33260fdf45b00 Mon Sep 17 00:00:00 2001 From: Ian Wright Date: Wed, 8 Mar 2023 13:39:49 +0000 Subject: [PATCH 033/113] more definitions --- docs/db.tex | 370 ++++++++++++++++++++++++++++++++-------------------- 1 file changed, 229 insertions(+), 141 deletions(-) diff --git a/docs/db.tex b/docs/db.tex index bc97fd3..a477df6 100644 --- a/docs/db.tex +++ b/docs/db.tex @@ -7,14 +7,15 @@ \usepackage{hyperref} \usepackage{url} +\usepackage{amsthm} -\title{Learning boolean functions with $\partial\mathbb{B}$ nets} +\title{$\partial\mathbb{B}$ nets: learning boolean functions with\\backpropagation} % Authors must not appear in the submitted version. They should be hidden % as long as the \iclrfinalcopy macro remains commented out below. % Non-anonymous submissions will be rejected without review. -\author{Ian Wright \thanks{GitHub, z80coder@github.com} +\author{Ian Wright \thanks{GitHub, Z80coder@github.com} } % The \author macro works with any number of authors. There are two commands @@ -28,20 +29,152 @@ \newcommand{\fix}{\marginpar{FIX}} \newcommand{\new}{\marginpar{NEW}} +\newtheorem{theorem}{Theorem} +\newtheorem*{definition}{Definition} +\newtheorem{prop}{Proposition} + + \iclrfinalcopy % Uncomment for camera-ready version, but NOT for submission. \begin{document} - \maketitle \begin{abstract} - $\partial\mathbb{B}$ nets are real-valued, differentiable neural networks that can be `hardened' to generate boolean functions with identical semantics. Existing approaches to neural network binarization lose accuracy; in contrast, training a $\partial\mathbb{B}$ net with backpropagation yields a compact and interpretable boolean function with identical accuracy. Experiments demonstrate that $\partial\mathbb{B}$ nets achieve competitive performance on standard machine learning problems. + $\partial\mathbb{B}$ nets are real-valued, differentiable neural networks + trained by backpropagation that yield boolean functions with identical accuracy. + Existing approaches to neural network binarization lose accuracy. In contrast, + $\partial\mathbb{B}$ nets `harden' to boolean functions with identical semantics. Experiments demonstrate that $\partial\mathbb{B}$ nets achieve competitive accuracy on standard machine learning problems yet are significantly more compact and interpretable. \end{abstract} \section{Introduction} -Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum. - +\section{Related work} + +\section{$\partial\mathbb{B}$ nets} + +\begin{definition}[Soft-bits and hard-bits] +A {\em soft-bit} is a real value in the range $[0,1]$ and a {\em hard-bit} is a boolean value in the set $\{0,1\}$. A soft-bit, $x$, is {\em high} if $x>1/2$, otherwise it is {\em low}. +\end{definition} + +\begin{definition}[Hardening] +The {\em hardening} function, $h(x_{1}, \dots, x_{n}) = [f(x_{1}), \dots, f(x_{n})]$, converts soft-bits to hard-bits, where +\begin{equation*} +f(x) = +\begin{cases} +1 & \text{if } x > 1/2 \\ +0 & \text{otherwise.} +\end{cases} +\end{equation*} +\end{definition} + +\begin{definition}[Hard-equivalence] + A function, $f: [0,1]^n \rightarrow [0,1]^m$, is {\em hard-equivalent} to a boolean function, $g: \{1,0\}^n \rightarrow \{1,0\}^m$, if + $h(f(h({\bf x}))) = g(h({\bf x}))$ for all ${\bf x} \in [0,1]^{n}$. +\end{definition} + +\subsection{Differentiable boolean functions} + +Weights are trainable soft-bits. A high weight implies the corresponding operation is masked out and therefore inactive. + +\begin{definition} +$\partial\text{NOT}: [0, 1]^{2} \rightarrow [0,1]$ is the function, + \begin{equation*} + \partial\text{NOT}(w, x) = 1 - w + x (2w - 1)\text{,} + \end{equation*} +where $w$ is a weight and $x$ is a soft-bit value. +\end{definition} + +\begin{prop}\label{prop:not} + $\partial${NOT} is hard-equivalent to the boolean function +$\neg (x \oplus w)$. +\end{prop} + +If the weight is high then $\partial${NOT} is hard-equivalent to the boolean identity function; otherwise it is hard-equivalent to $\neg$. In consequence, we can learn to logically not, or simply pass through, the input value $x$. + +\begin{definition} +$\partial\text{AND}: [0,1]^{2} \rightarrow [0,1]$ is the function, + \begin{equation*} + \partial\text{AND}(x, y) = + \begin{cases} + 1/2 + 1/2(x + y)(m - 1/2) & \text{if } 2m > 1 \\ + m + 1/2(x + y)(1/2 - m) & \text{otherwise,} + \end{cases} + \end{equation*} + where $m=\min(x,y)$, and $x$ and $y$ are soft-bit values. +\end{definition} + +\begin{prop}\label{prop:and} + $\partial${AND} is hard-equivalent to the boolean function $x \wedge y$. +\end{prop} + +\begin{definition} +$\partial\text{OR}: [0,1]^{2} \rightarrow [0,1]$ is the function, + \begin{equation*} + \partial\text{OR}(x, y) = + \begin{cases} + 1/2 + 1/2(x + y)(m - 1/2) & \text{if } 2m > 1 \\ + m + 1/2(x + y)(1/2 - m) & \text{otherwise,} + \end{cases} + \end{equation*} + where $m=\max(x,y)$, and $x$ and $y$ are soft-bit values. +\end{definition} + +\begin{prop}\label{prop:or} + $\partial${OR} is hard-equivalent to the boolean function $x \vee y$. +\end{prop} + +\begin{definition} +$\partial\text{IMPLIES}: [0,1]^{2} \rightarrow [0,1]$ is the function, + \begin{equation*} + \partial\text{IMPLIES}(w, x) = \partial\text{OR}(x, 1-w)\text{,} + \end{equation*} +where $w$ is a weight and $x$ is a soft-bit value. +\end{definition} + +\begin{prop}\label{prop:implies} + $\partial${IMPLIES} is hard-equivalent to the boolean function $w \Rightarrow x$. +\end{prop} + +Define $\operatorname{majority-index}: \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{> 0}$ as $\operatorname{majority-index}(n) = 1 + \lfloor \frac{n-1}{2}\rfloor$. + +Define $\operatorname{select}: [0,1]^n \times {1, 2, \ldots, n} \rightarrow [0,1]$ as $\operatorname{select}({\bf x}, i) = x_{i}$. + +Define $\operatorname{majority-bit}: [0,1]^n \rightarrow [0,1]$ as $$\operatorname{majority-bit}({\bf x}) = \operatorname{select}( \operatorname{sort}({\bf x}), \operatorname{majority-index}(\lvert{\bf x}\rvert))\text{,}$$ where $\operatorname{sort}$ sorts the elements of a vector in ascending order. + +Define $\operatorname{majority-delta}: [0,1]^n \rightarrow [0,1]$ as +$$\operatorname{majority-delta}({\bf x}) = \bar{\bf x} \times \left| \operatorname{majority-bit}({\bf x}) - 1/2\right|\text{.}$$ + +\begin{definition} +$\partial\text{MAJORITY}: [0,1]^{n} \rightarrow [0,1]$ is the function, + \begin{equation*} + \partial\text{MAJORITY}({\bf x}) = + \begin{cases} + 1/2 + \delta & \text{if } m > 1/2 \\ + m + \delta & \text{otherwise,} + \end{cases} + \end{equation*} + where ${\bf x}$ is a vector of soft-bits, $m = \operatorname{majority-bit}({\bf x})$ and $\delta = \operatorname{majority-delta}({\bf x})$. +\end{definition} + +\begin{prop}\label{prop:majority} + $\partial${MAJORITY} is hard-equivalent to the boolean majority function, +\begin{equation*} +\text{MAJ}(x_{1}, \dots, x_{n}) = + \begin{cases} + 1 & \text{if } \sum_{i=1}^{n} x_{i} \geq n/2 \\ + 0 & \text{otherwise.} +\end{cases} +\end{equation*} +\end{prop} + + +\subsection{Logical layers} + + + +\section{Experiments} + +\section{Conclusion} \begin{figure}[h] \begin{center} @@ -51,139 +184,6 @@ \section{Introduction} \caption{Sample figure caption.} \end{figure} -\section{Tables} - -\begin{table}[h] -\caption{Sample table title} -\label{sample-table} -\begin{center} -\begin{tabular}{ll} -\multicolumn{1}{c}{\bf PART} &\multicolumn{1}{c}{\bf DESCRIPTION} -\\ \hline \\ -Dendrite &Input terminal \\ -Axon &Output terminal \\ -Soma &Cell body (contains cell nucleus) \\ -\end{tabular} -\end{center} -\end{table} - -\section{Default Notation} - - -\centerline{\bf Numbers and Arrays} -\bgroup -\def\arraystretch{1.5} -\begin{tabular}{p{1in}p{3.25in}} -$\displaystyle a$ & A scalar (integer or real)\\ -$\displaystyle \va$ & A vector\\ -$\displaystyle \mA$ & A matrix\\ -$\displaystyle \tA$ & A tensor\\ -$\displaystyle \mI_n$ & Identity matrix with $n$ rows and $n$ columns\\ -$\displaystyle \mI$ & Identity matrix with dimensionality implied by context\\ -$\displaystyle \ve^{(i)}$ & Standard basis vector $[0,\dots,0,1,0,\dots,0]$ with a 1 at position $i$\\ -$\displaystyle \text{diag}(\va)$ & A square, diagonal matrix with diagonal entries given by $\va$\\ -$\displaystyle \ra$ & A scalar random variable\\ -$\displaystyle \rva$ & A vector-valued random variable\\ -$\displaystyle \rmA$ & A matrix-valued random variable\\ -\end{tabular} -\egroup -\vspace{0.25cm} - -\centerline{\bf Sets and Graphs} -\bgroup -\def\arraystretch{1.5} - -\begin{tabular}{p{1.25in}p{3.25in}} -$\displaystyle \sA$ & A set\\ -$\displaystyle \R$ & The set of real numbers \\ -$\displaystyle \{0, 1\}$ & The set containing 0 and 1 \\ -$\displaystyle \{0, 1, \dots, n \}$ & The set of all integers between $0$ and $n$\\ -$\displaystyle [a, b]$ & The real interval including $a$ and $b$\\ -$\displaystyle (a, b]$ & The real interval excluding $a$ but including $b$\\ -$\displaystyle \sA \backslash \sB$ & Set subtraction, i.e., the set containing the elements of $\sA$ that are not in $\sB$\\ -$\displaystyle \gG$ & A graph\\ -$\displaystyle \parents_\gG(\ervx_i)$ & The parents of $\ervx_i$ in $\gG$ -\end{tabular} -\vspace{0.25cm} - - -\centerline{\bf Indexing} -\bgroup -\def\arraystretch{1.5} - -\begin{tabular}{p{1.25in}p{3.25in}} -$\displaystyle \eva_i$ & Element $i$ of vector $\va$, with indexing starting at 1 \\ -$\displaystyle \eva_{-i}$ & All elements of vector $\va$ except for element $i$ \\ -$\displaystyle \emA_{i,j}$ & Element $i, j$ of matrix $\mA$ \\ -$\displaystyle \mA_{i, :}$ & Row $i$ of matrix $\mA$ \\ -$\displaystyle \mA_{:, i}$ & Column $i$ of matrix $\mA$ \\ -$\displaystyle \etA_{i, j, k}$ & Element $(i, j, k)$ of a 3-D tensor $\tA$\\ -$\displaystyle \tA_{:, :, i}$ & 2-D slice of a 3-D tensor\\ -$\displaystyle \erva_i$ & Element $i$ of the random vector $\rva$ \\ -\end{tabular} -\egroup -\vspace{0.25cm} - - -\centerline{\bf Calculus} -\bgroup -\def\arraystretch{1.5} -\begin{tabular}{p{1.25in}p{3.25in}} -% NOTE: the [2ex] on the next line adds extra height to that row of the table. -% Without that command, the fraction on the first line is too tall and collides -% with the fraction on the second line. -$\displaystyle\frac{d y} {d x}$ & Derivative of $y$ with respect to $x$\\ [2ex] -$\displaystyle \frac{\partial y} {\partial x} $ & Partial derivative of $y$ with respect to $x$ \\ -$\displaystyle \nabla_\vx y $ & Gradient of $y$ with respect to $\vx$ \\ -$\displaystyle \nabla_\mX y $ & Matrix derivatives of $y$ with respect to $\mX$ \\ -$\displaystyle \nabla_\tX y $ & Tensor containing derivatives of $y$ with respect to $\tX$ \\ -$\displaystyle \frac{\partial f}{\partial \vx} $ & Jacobian matrix $\mJ \in \R^{m\times n}$ of $f: \R^n \rightarrow \R^m$\\ -$\displaystyle \nabla_\vx^2 f(\vx)\text{ or }\mH( f)(\vx)$ & The Hessian matrix of $f$ at input point $\vx$\\ -$\displaystyle \int f(\vx) d\vx $ & Definite integral over the entire domain of $\vx$ \\ -$\displaystyle \int_\sS f(\vx) d\vx$ & Definite integral with respect to $\vx$ over the set $\sS$ \\ -\end{tabular} -\egroup -\vspace{0.25cm} - -\centerline{\bf Probability and Information Theory} -\bgroup -\def\arraystretch{1.5} -\begin{tabular}{p{1.25in}p{3.25in}} -$\displaystyle P(\ra)$ & A probability distribution over a discrete variable\\ -$\displaystyle p(\ra)$ & A probability distribution over a continuous variable, or over -a variable whose type has not been specified\\ -$\displaystyle \ra \sim P$ & Random variable $\ra$ has distribution $P$\\% so thing on left of \sim should always be a random variable, with name beginning with \r -$\displaystyle \E_{\rx\sim P} [ f(x) ]\text{ or } \E f(x)$ & Expectation of $f(x)$ with respect to $P(\rx)$ \\ -$\displaystyle \Var(f(x)) $ & Variance of $f(x)$ under $P(\rx)$ \\ -$\displaystyle \Cov(f(x),g(x)) $ & Covariance of $f(x)$ and $g(x)$ under $P(\rx)$\\ -$\displaystyle H(\rx) $ & Shannon entropy of the random variable $\rx$\\ -$\displaystyle \KL ( P \Vert Q ) $ & Kullback-Leibler divergence of P and Q \\ -$\displaystyle \mathcal{N} ( \vx ; \vmu , \mSigma)$ & Gaussian distribution % -over $\vx$ with mean $\vmu$ and covariance $\mSigma$ \\ -\end{tabular} -\egroup -\vspace{0.25cm} - -\centerline{\bf Functions} -\bgroup -\def\arraystretch{1.5} -\begin{tabular}{p{1.25in}p{3.25in}} -$\displaystyle f: \sA \rightarrow \sB$ & The function $f$ with domain $\sA$ and range $\sB$\\ -$\displaystyle f \circ g $ & Composition of the functions $f$ and $g$ \\ - $\displaystyle f(\vx ; \vtheta) $ & A function of $\vx$ parametrized by $\vtheta$. - (Sometimes we write $f(\vx)$ and omit the argument $\vtheta$ to lighten notation) \\ -$\displaystyle \log x$ & Natural logarithm of $x$ \\ -$\displaystyle \sigma(x)$ & Logistic sigmoid, $\displaystyle \frac{1} {1 + \exp(-x)}$ \\ -$\displaystyle \zeta(x)$ & Softplus, $\log(1 + \exp(x))$ \\ -$\displaystyle || \vx ||_p $ & $\normlp$ norm of $\vx$ \\ -$\displaystyle || \vx || $ & $\normltwo$ norm of $\vx$ \\ -$\displaystyle x^+$ & Positive part of $x$, i.e., $\max(0,x)$\\ -$\displaystyle \1_\mathrm{condition}$ & is 1 if the condition is true, 0 otherwise\\ -\end{tabular} -\egroup -\vspace{0.25cm} - - \subsubsection*{Acknowledgments} Use unnumbered third level headings for the acknowledgments. All @@ -194,7 +194,95 @@ \subsubsection*{Acknowledgments} \bibliographystyle{iclr2021_conference} \appendix -\section{Appendix} -You may include other additional sections here. + +\section*{Appendix} + +\section{Proofs} + +\subsection{Proof of proposition~\ref{prop:not}} + +\begin{proof} + Table \ref{not-table} is the truth table of the boolean function $\neg (x \oplus w)$. + \begin{table} + \begin{center} + \begin{tabular}{lll} + \multicolumn{1}{c}{$h(w)$} &\multicolumn{1}{c}{$h(x)$} &\multicolumn{1}{c}{$\partial\text{NOT}(h(w), h(x))$} + \\ \hline \\ + 0 & 0 & 1\\ + 1 & 0 & 0\\ + 0 & 1 & 0\\ + 1 & 1 & 1\\ + \end{tabular} + \end{center} + \caption{$\partial${NOT} is hard-equivalent to $\neg (x \oplus w)$.}\label{not-table} + + \end{table} +\end{proof} + +\subsection{Proof of proposition~\ref{prop:and}} + +\begin{proof} + Table \ref{and-table} is the truth table of the boolean function $x \wedge y$. + \begin{table} + \begin{center} + \begin{tabular}{llll} + \multicolumn{1}{c}{$h(x)$} &\multicolumn{1}{c}{$h(y)$} &\multicolumn{1}{c}{$\partial\text{AND}(h(x), h(y))$} &\multicolumn{1}{c}{$h(\partial\text{AND}(h(x), h(y)))$} + \\ \hline \\ + 1 & 1 & 1 & 1\\ + 1 & 0 & 1/4 & 0\\ + 0 & 1 & 1/4 & 0\\ + 0 & 0 & 0 & 0\\ + \end{tabular} + \end{center} + \caption{$\partial${AND} is hard-equivalent to $x \wedge y$.}\label{and-table} + + \end{table} +\end{proof} + +\subsection{Proof of proposition~\ref{prop:or}} + +\begin{proof} + Table \ref{or-table} is the truth table of the boolean function $x \vee y$. + \begin{table} + \begin{center} + \begin{tabular}{llll} + \multicolumn{1}{c}{$h(x)$} &\multicolumn{1}{c}{$h(y)$} &\multicolumn{1}{c}{$\partial\text{OR}(h(x), h(y))$} &\multicolumn{1}{c}{$h(\partial\text{OR}(h(x), h(y)))$} + \\ \hline \\ + 1 & 1 & 1 & 1\\ + 1 & 0 & 3/4 & 1\\ + 0 & 1 & 3/4 & 1\\ + 0 & 0 & 0 & 0\\ + \end{tabular} + \end{center} + \caption{$\partial${OR} is hard-equivalent to $x \vee y$.}\label{or-table} + + \end{table} +\end{proof} + +\subsection{Proof of proposition~\ref{prop:implies}} + +\begin{proof} + Table \ref{implies-table} is the truth table of the boolean function $x \Rightarrow y$. + \begin{table} + \begin{center} + \begin{tabular}{llll} + \multicolumn{1}{c}{$h(w)$} &\multicolumn{1}{c}{$h(x)$} &\multicolumn{1}{c}{$\partial\text{IMPLIES}(h(w), h(x))$} &\multicolumn{1}{c}{$h(\partial\text{IMPLIES}(h(w), h(x)))$} + \\ \hline \\ + 1 & 1 & 3/4 & 1\\ + 1 & 0 & 0 & 0\\ + 0 & 1 & 1 & 1\\ + 0 & 0 & 3/4 & 1\\ + \end{tabular} + \end{center} + \caption{$\partial${IMPLIES} is hard-equivalent to $w \Rightarrow x$.}\label{implies-table} + + \end{table} +\end{proof} + +\subsection{Proof of proposition~\ref{prop:majority}} + +\begin{proof} +todo +\end{proof} \end{document} From 2ed7989a41f0cd7be2da1de38877b3d3a695fced Mon Sep 17 00:00:00 2001 From: Ian Wright Date: Thu, 9 Mar 2023 09:30:14 +0000 Subject: [PATCH 034/113] more concise definitions --- docs/db.tex | 170 ++++++++++++++++++++++++++-------------------------- 1 file changed, 86 insertions(+), 84 deletions(-) diff --git a/docs/db.tex b/docs/db.tex index a477df6..6b4ff2c 100644 --- a/docs/db.tex +++ b/docs/db.tex @@ -32,6 +32,7 @@ \newtheorem{theorem}{Theorem} \newtheorem*{definition}{Definition} \newtheorem{prop}{Proposition} +\newtheorem{lemma}{Lemma} \iclrfinalcopy % Uncomment for camera-ready version, but NOT for submission. @@ -53,7 +54,7 @@ \section{Related work} \section{$\partial\mathbb{B}$ nets} \begin{definition}[Soft-bits and hard-bits] -A {\em soft-bit} is a real value in the range $[0,1]$ and a {\em hard-bit} is a boolean value in the set $\{0,1\}$. A soft-bit, $x$, is {\em high} if $x>1/2$, otherwise it is {\em low}. +A {\em soft-bit} is a real value in the range $[0,1]$ and a {\em hard-bit} is a boolean value from the set $\{0,1\}$. A soft-bit, $x$, is {\em high} if $x>1/2$, otherwise it is {\em low}. \end{definition} \begin{definition}[Hardening] @@ -76,96 +77,75 @@ \subsection{Differentiable boolean functions} Weights are trainable soft-bits. A high weight implies the corresponding operation is masked out and therefore inactive. -\begin{definition} -$\partial\text{NOT}: [0, 1]^{2} \rightarrow [0,1]$ is the function, +Define \begin{equation*} - \partial\text{NOT}(w, x) = 1 - w + x (2w - 1)\text{,} + \begin{aligned} + \partial\text{NOT}: [0, 1]^{2} &\to [0,1], \\ + (w, x) &\mapsto 1 - w + x (2w - 1)\text{,} + \end{aligned} \end{equation*} where $w$ is a weight and $x$ is a soft-bit value. -\end{definition} - -\begin{prop}\label{prop:not} - $\partial${NOT} is hard-equivalent to the boolean function -$\neg (x \oplus w)$. -\end{prop} +$\partial${NOT} is hard-equivalent to the boolean function $\neg(x \oplus w)$ (proposition \ref{prop:not}). If the weight is high then $\partial${NOT} is hard-equivalent to the boolean identity function; otherwise it is hard-equivalent to $\neg$. In consequence, we can learn to logically not, or simply pass through, the input value $x$. -If the weight is high then $\partial${NOT} is hard-equivalent to the boolean identity function; otherwise it is hard-equivalent to $\neg$. In consequence, we can learn to logically not, or simply pass through, the input value $x$. - -\begin{definition} -$\partial\text{AND}: [0,1]^{2} \rightarrow [0,1]$ is the function, - \begin{equation*} - \partial\text{AND}(x, y) = +Define +\begin{equation*} +\begin{aligned} +\partial\text{AND}: [0,1]^{2} &\to [0,1], \\ + (x, y) &\mapsto \begin{cases} 1/2 + 1/2(x + y)(m - 1/2) & \text{if } 2m > 1 \\ m + 1/2(x + y)(1/2 - m) & \text{otherwise,} \end{cases} - \end{equation*} - where $m=\min(x,y)$, and $x$ and $y$ are soft-bit values. -\end{definition} - -\begin{prop}\label{prop:and} - $\partial${AND} is hard-equivalent to the boolean function $x \wedge y$. -\end{prop} +\end{aligned} +\end{equation*} +where $m=\min(x,y)$, and $x$ and $y$ are soft-bit values. +$\partial${AND} is hard-equivalent to the boolean function $x \wedge y$ (proposition \ref{prop:and}). -\begin{definition} -$\partial\text{OR}: [0,1]^{2} \rightarrow [0,1]$ is the function, - \begin{equation*} - \partial\text{OR}(x, y) = - \begin{cases} +Define +\begin{equation*} +\begin{aligned} +\partial\text{OR}: [0,1]^{2} &\to [0,1], \\ +(x, y) &\mapsto +\begin{cases} 1/2 + 1/2(x + y)(m - 1/2) & \text{if } 2m > 1 \\ - m + 1/2(x + y)(1/2 - m) & \text{otherwise,} - \end{cases} - \end{equation*} - where $m=\max(x,y)$, and $x$ and $y$ are soft-bit values. -\end{definition} - -\begin{prop}\label{prop:or} - $\partial${OR} is hard-equivalent to the boolean function $x \vee y$. -\end{prop} +m + 1/2(x + y)(1/2 - m) & \text{otherwise,} +\end{cases} +\end{aligned} +\end{equation*} +where $m=\max(x,y)$, and $x$ and $y$ are soft-bit values. +$\partial${OR} is hard-equivalent to the boolean function $x \vee y$ (proposition \ref{prop:or}). -\begin{definition} -$\partial\text{IMPLIES}: [0,1]^{2} \rightarrow [0,1]$ is the function, - \begin{equation*} - \partial\text{IMPLIES}(w, x) = \partial\text{OR}(x, 1-w)\text{,} - \end{equation*} +Define +\begin{equation*} +\begin{aligned} +\partial\text{IMPLIES}: [0,1]^{2} &\to [0,1],\\ +(w, x) &\mapsto \partial\text{OR}(x, 1-w)\text{,} +\end{aligned} +\end{equation*} where $w$ is a weight and $x$ is a soft-bit value. -\end{definition} - -\begin{prop}\label{prop:implies} - $\partial${IMPLIES} is hard-equivalent to the boolean function $w \Rightarrow x$. -\end{prop} +$\partial${IMPLIES} is hard-equivalent to the boolean function $w \Rightarrow x$ (proposition \ref{prop:implies}). -Define $\operatorname{majority-index}: \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{> 0}$ as $\operatorname{majority-index}(n) = 1 + \lfloor \frac{n-1}{2}\rfloor$. +Define $\operatorname{majority-index}: \mathbb{Z}_{>0} \to \mathbb{Z}_{> 0}$ as $n \mapsto 1 + \lfloor \frac{n-1}{2}\rfloor$. -Define $\operatorname{select}: [0,1]^n \times {1, 2, \ldots, n} \rightarrow [0,1]$ as $\operatorname{select}({\bf x}, i) = x_{i}$. +Define $\operatorname{select}: [0,1]^n \times {1, 2, \ldots, n} \to [0,1]$ as $({\bf x}, i) \mapsto x_{i}$. -Define $\operatorname{majority-bit}: [0,1]^n \rightarrow [0,1]$ as $$\operatorname{majority-bit}({\bf x}) = \operatorname{select}( \operatorname{sort}({\bf x}), \operatorname{majority-index}(\lvert{\bf x}\rvert))\text{,}$$ where $\operatorname{sort}$ sorts the elements of a vector in ascending order. +Define $\operatorname{majority-bit}: [0,1]^n \to [0,1]$ as ${\bf x} \mapsto \operatorname{select}( \operatorname{sort}({\bf x}), \operatorname{majority-index}(\lvert{\bf x}\rvert))$, where $\operatorname{sort}({\bf x})$ sorts the elements of ${\bf x}$ in ascending order. -Define $\operatorname{majority-delta}: [0,1]^n \rightarrow [0,1]$ as -$$\operatorname{majority-delta}({\bf x}) = \bar{\bf x} \times \left| \operatorname{majority-bit}({\bf x}) - 1/2\right|\text{.}$$ +Define $\operatorname{majority-delta}: [0,1]^n \to [0,1]$ as +${\bf x} \mapsto \bar{\bf x} \times \left| \operatorname{majority-bit}({\bf x}) - 1/2\right|$. -\begin{definition} -$\partial\text{MAJORITY}: [0,1]^{n} \rightarrow [0,1]$ is the function, - \begin{equation*} - \partial\text{MAJORITY}({\bf x}) = +\begin{equation*} +\begin{aligned} +\partial\text{MAJORITY}: [0,1]^{n} &\to [0,1], \\ +{\bf x} &\mapsto \begin{cases} 1/2 + \delta & \text{if } m > 1/2 \\ m + \delta & \text{otherwise,} \end{cases} - \end{equation*} - where ${\bf x}$ is a vector of soft-bits, $m = \operatorname{majority-bit}({\bf x})$ and $\delta = \operatorname{majority-delta}({\bf x})$. -\end{definition} - -\begin{prop}\label{prop:majority} - $\partial${MAJORITY} is hard-equivalent to the boolean majority function, -\begin{equation*} -\text{MAJ}(x_{1}, \dots, x_{n}) = - \begin{cases} - 1 & \text{if } \sum_{i=1}^{n} x_{i} \geq n/2 \\ - 0 & \text{otherwise.} -\end{cases} +\end{aligned} \end{equation*} -\end{prop} +where ${\bf x}$ is a vector of soft-bits, $m = \operatorname{majority-bit}({\bf x})$ and $\delta = \operatorname{majority-delta}({\bf x})$. +$\partial${MAJORITY} is hard-equivalent to the boolean majority function (proposition \ref{prop:majority}). \subsection{Logical layers} @@ -176,13 +156,13 @@ \section{Experiments} \section{Conclusion} -\begin{figure}[h] -\begin{center} +%\begin{figure}[h] +%\begin{center} %\framebox[4.0in]{$\;$} -\fbox{\rule[-.5cm]{0cm}{4cm} \rule[-.5cm]{4cm}{0cm}} -\end{center} -\caption{Sample figure caption.} -\end{figure} +%\fbox{\rule[-.5cm]{0cm}{4cm} \rule[-.5cm]{4cm}{0cm}} +%\end{center} +%\caption{Sample figure caption.} +%\end{figure} \subsubsection*{Acknowledgments} @@ -199,8 +179,9 @@ \section*{Appendix} \section{Proofs} -\subsection{Proof of proposition~\ref{prop:not}} - +\begin{prop}\label{prop:not} + $\partial${NOT} is hard-equivalent to the boolean function + $\neg (x \oplus w)$. \begin{proof} Table \ref{not-table} is the truth table of the boolean function $\neg (x \oplus w)$. \begin{table} @@ -218,9 +199,11 @@ \subsection{Proof of proposition~\ref{prop:not}} \end{table} \end{proof} +\end{prop} -\subsection{Proof of proposition~\ref{prop:and}} +\begin{prop}\label{prop:and} + $\partial${AND} is hard-equivalent to the boolean function $x \wedge y$. \begin{proof} Table \ref{and-table} is the truth table of the boolean function $x \wedge y$. \begin{table} @@ -238,9 +221,10 @@ \subsection{Proof of proposition~\ref{prop:and}} \end{table} \end{proof} +\end{prop} -\subsection{Proof of proposition~\ref{prop:or}} - +\begin{prop}\label{prop:or} + $\partial${OR} is hard-equivalent to the boolean function $x \vee y$. \begin{proof} Table \ref{or-table} is the truth table of the boolean function $x \vee y$. \begin{table} @@ -258,9 +242,10 @@ \subsection{Proof of proposition~\ref{prop:or}} \end{table} \end{proof} +\end{prop} -\subsection{Proof of proposition~\ref{prop:implies}} - +\begin{prop}\label{prop:implies} + $\partial${IMPLIES} is hard-equivalent to the boolean function $w \Rightarrow x$. \begin{proof} Table \ref{implies-table} is the truth table of the boolean function $x \Rightarrow y$. \begin{table} @@ -278,11 +263,28 @@ \subsection{Proof of proposition~\ref{prop:implies}} \end{table} \end{proof} +\end{prop} -\subsection{Proof of proposition~\ref{prop:majority}} +\begin{lemma} +$\operatorname{majority-bit}$ is hard-equivalent to the boolean majority function. +\begin{proof} + The boolean majority function, +\begin{equation*} +\text{MAJ}(x_{1}, \dots, x_{n}) = +\begin{cases} +1 & \text{if } \sum_{i=1}^{n} x_{i} \geq n/2 \\ +0 & \text{otherwise.} +\end{cases} +\end{equation*} +\end{proof} +\end{lemma} +\begin{prop}\label{prop:majority} + $\partial${MAJORITY} is hard-equivalent to the boolean majority function. \begin{proof} -todo + todo \end{proof} +\end{prop} + \end{document} From fa327a366f399ef9861f7f2452531371e97b36af Mon Sep 17 00:00:00 2001 From: Ian Wright Date: Thu, 9 Mar 2023 13:03:58 +0000 Subject: [PATCH 035/113] more --- docs/db.tex | 63 +- docs/noisy-xor-architecture.png | Bin 0 -> 32015 bytes docs/proofs.nb | 1926 +++++++++++++++++++++++++++++++ 3 files changed, 1985 insertions(+), 4 deletions(-) create mode 100644 docs/noisy-xor-architecture.png create mode 100644 docs/proofs.nb diff --git a/docs/db.tex b/docs/db.tex index 6b4ff2c..0a3da25 100644 --- a/docs/db.tex +++ b/docs/db.tex @@ -8,6 +8,8 @@ \usepackage{hyperref} \usepackage{url} \usepackage{amsthm} +\usepackage{graphicx} +\usepackage{comment} \title{$\partial\mathbb{B}$ nets: learning boolean functions with\\backpropagation} @@ -42,9 +44,8 @@ \begin{abstract} $\partial\mathbb{B}$ nets are real-valued, differentiable neural networks - trained by backpropagation that yield boolean functions with identical accuracy. - Existing approaches to neural network binarization lose accuracy. In contrast, - $\partial\mathbb{B}$ nets `harden' to boolean functions with identical semantics. Experiments demonstrate that $\partial\mathbb{B}$ nets achieve competitive accuracy on standard machine learning problems yet are significantly more compact and interpretable. + trained by backpropagation that learn discrete boolean functions. + $\partial\mathbb{B}$ nets, once trained, `harden' to boolean functions with identical semantics and therefore, unlike existing approaches to neural network binarization, retain their accuracy. Experiments demonstrate that $\partial\mathbb{B}$ nets achieve competitive performance on standard machine learning problems yet are significantly more compact (due to 1-bit weights) and interpretable (due to the logical nature of the learnt function). \end{abstract} \section{Introduction} @@ -134,6 +135,7 @@ \subsection{Differentiable boolean functions} Define $\operatorname{majority-delta}: [0,1]^n \to [0,1]$ as ${\bf x} \mapsto \bar{\bf x} \times \left| \operatorname{majority-bit}({\bf x}) - 1/2\right|$. +Define \begin{equation*} \begin{aligned} \partial\text{MAJORITY}: [0,1]^{n} &\to [0,1], \\ @@ -147,13 +149,56 @@ \subsection{Differentiable boolean functions} where ${\bf x}$ is a vector of soft-bits, $m = \operatorname{majority-bit}({\bf x})$ and $\delta = \operatorname{majority-delta}({\bf x})$. $\partial${MAJORITY} is hard-equivalent to the boolean majority function (proposition \ref{prop:majority}). - \subsection{Logical layers} +Define +\begin{equation*} +\begin{aligned} +\partial\text{NOT-LAYER}: [0,1]^{n \times m} \times [0,1]^{m} &\to [0,1]^{n \times m}, \\ +({\bf W}, {\bf x}) &\mapsto +\begin{bmatrix} +\partial\text{NOT}(w_{1,1}, x_{1}) & \dots & \partial\text{NOT}(w_{1,m}, x_{m}) \\ +\vdots & \ddots & \vdots \\ +\partial\text{NOT}(w_{n,1}, x_{1}) & \dots & \partial\text{NOT}(w_{n,m}, x_{m}) +\end{bmatrix} +\end{aligned} +\end{equation*} +where ${\bf W}$ is a matrix of weights and ${\bf x}$ is a vector of soft-bits. + +%[\partial\text{NOT}({\bf W}_{1}, {\bf x}), \dots, \partial\text{NOT}({\bf W}_{n}, {\bf x})] + +Define +\begin{equation*} +\begin{aligned} +\partial\text{AND-NEURON}: [0,1]^{n} \times [0,1]^{n} &\to [0,1], \\ +({\bf w}, {\bf x}) &\mapsto \min(\partial\text{IMPLIES}(w_{1}, x_{1}), \dots, \partial\text{IMPLIES}(w_{n}, x_{n}))\text{,} +\end{aligned} +\end{equation*} +where ${\bf w}$ is vector of weights and ${\bf x}$ is a vector of soft-bits. A single AND neuron maps $n$ soft-bit inputs to a single soft-bit value. A $\partial\text{AND-LAYER}$ of $n$ neurons maps $m$ soft-bit inputs to $n$ soft-bit outputs. + +Define +\begin{equation*} +\begin{aligned} +\partial\text{OR-NEURON}: [0,1]^{n} \times [0,1]^{n} &\to [0,1], \\ +({\bf w}, {\bf x}) &\mapsto \max(\partial\text{AND}(w_{1}, x_{1}), \dots, \partial\text{AND}(w_{n}, x_{n}))\text{.} +\end{aligned} +\end{equation*} +A single OR neuron maps $n$ soft-bit inputs to a single soft-bit value. A $\partial\text{OR-LAYER}$ of $n$ neurons maps $m$ soft-bit inputs to $n$ soft-bit outputs. \section{Experiments} +\subsection{Noisy XOR} + +\begin{figure}[h] + \centering + \includegraphics[width=0.8\textwidth]{noisy-xor-architecture.png} + \caption{todo} + \label{fig:noisy-xor-architecture} +\end{figure} + +todo: weight initialization + \section{Conclusion} %\begin{figure}[h] @@ -288,3 +333,13 @@ \section{Proofs} \end{document} + +\begin{comment} +Define +\begin{equation*} +\begin{aligned} +\partial\text{AND-LAYER}: [0,1]^{n \times m} \times [0,1]^{m} &\to [0,1]^{n}, \\ +({\bf W}, {\bf x}) &\mapsto [\partial\text{AND-NEURON}({\bf W}_{1}, {\bf x}), \dots, \partial\text{AND-NEURON}({\bf W}_{n}, {\bf x})] +\end{aligned} +\end{equation*} +\end{comment} diff --git a/docs/noisy-xor-architecture.png b/docs/noisy-xor-architecture.png new file mode 100644 index 0000000000000000000000000000000000000000..12e6579b2d73299c988be407134351aef0fe4537 GIT binary patch literal 32015 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++++++++++++++++++++++++-------- docs/noisy-xor-architecture.png | Bin 32015 -> 31110 bytes 2 files changed, 30 insertions(+), 9 deletions(-) diff --git a/docs/db.tex b/docs/db.tex index 0a3da25..9f39cd6 100644 --- a/docs/db.tex +++ b/docs/db.tex @@ -193,21 +193,42 @@ \subsection{Noisy XOR} \begin{figure}[h] \centering \includegraphics[width=0.8\textwidth]{noisy-xor-architecture.png} - \caption{todo} + \caption{$\partial\mathbb{B}$ net for the noisy-xor problem.} \label{fig:noisy-xor-architecture} \end{figure} todo: weight initialization -\section{Conclusion} +\begin{table}[] + \centering + \begin{tabular}{llllll} + \multicolumn{1}{c}{} & \multicolumn{5}{c}{accuracy} \\ + \multicolumn{1}{l|}{} & \multicolumn{1}{l|}{mean} & \multicolumn{1}{l|}{5 \%ile} & \multicolumn{1}{l|}{95 \%ile} & \multicolumn{1}{l|}{min} & \multicolumn{1}{l|}{max} \\ \hline + \multicolumn{1}{|l|}{Tsetlin} & \multicolumn{1}{l|}{99.3 +/- 0.3} & \multicolumn{1}{l|}{95.9} & \multicolumn{1}{l|}{100.0} & \multicolumn{1}{l|}{91.6} & \multicolumn{1}{l|}{100.0} \\ \hline + \multicolumn{1}{|l|}{$\partial\mathbb{B}$} & \multicolumn{1}{l|}{\textbf{97.9 +/- 0.2}} & \multicolumn{1}{l|}{\textbf{95.4}} & \multicolumn{1}{l|}{\textbf{100.0}} & \multicolumn{1}{l|}{\textbf{93.6}} & \multicolumn{1}{l|}{\textbf{100.0}} \\ \hline + \multicolumn{1}{|l|}{neural network} & \multicolumn{1}{l|}{95.4 +/- 0.5} & \multicolumn{1}{l|}{90.1} & \multicolumn{1}{l|}{98.6} & \multicolumn{1}{l|}{88.2} & \multicolumn{1}{l|}{99.9} \\ \hline + \multicolumn{1}{|l|}{SVM} & \multicolumn{1}{l|}{58.0 +/- 0.3} & \multicolumn{1}{l|}{56.4} & \multicolumn{1}{l|}{59.2} & \multicolumn{1}{l|}{55.4} & \multicolumn{1}{l|}{66.5} \\ \hline + \multicolumn{1}{|l|}{naive Bayes} & \multicolumn{1}{l|}{49.8 +/- 0.2} & \multicolumn{1}{l|}{48.3} & \multicolumn{1}{l|}{51.0} & \multicolumn{1}{l|}{41.3} & \multicolumn{1}{l|}{52.7} \\ \hline + \multicolumn{1}{|l|}{logistic regression} & \multicolumn{1}{l|}{49.8 +/- 0.3} & \multicolumn{1}{l|}{47.8} & \multicolumn{1}{l|}{51.1} & \multicolumn{1}{l|}{41.1} & \multicolumn{1}{l|}{53.1} \\ \hline + \end{tabular} + \caption{Noisy-XOR results.} + \label{tab:noisy-xor-results} +\end{table} +\begin{comment} +""" +| Technique/Accuracy | Mean | 5 %ile | 95 %ile | Min | Max | +| ------------------- | -------------- | ------- | ------- | ------ | ------ | +| Tsetlin | 99.3 +/- 0.3 | 95.9 | 100.0 | 91.6 | 100.0 | +| dB | 97.9 +/- 0.2 | 95.4 | 100.0 | 93.6 | 100.0 | +| Neural network | 95.4 +/- 0.5 | 90.1 | 98.6 | 88.2 | 99.9 | +| SVM | 58.0 +/- 0.3 | 56.4 | 59.2 | 55.4 | 66.5 | +| Naive Bayes | 49.8 +/- 0.2 | 48.3 | 51.0 | 41.3 | 52.7 | +| Logistic regression | 49.8 +/- 0.3 | 47.8 | 51.1 | 41.1 | 53.1 | +Source: https://arxiv.org/pdf/1804.01508.pdf +""" +\end{comment} -%\begin{figure}[h] -%\begin{center} -%\framebox[4.0in]{$\;$} -%\fbox{\rule[-.5cm]{0cm}{4cm} \rule[-.5cm]{4cm}{0cm}} -%\end{center} -%\caption{Sample figure caption.} -%\end{figure} +\section{Conclusion} 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docs/db.tex | 28 +- docs/iclr2021_conference.bst | 1440 +++++++++++++++++++++++++++++++ docs/iclr2021_conference.sty | 5 +- docs/noisy-xor-architecture.png | Bin 31110 -> 32684 bytes 4 files changed, 1453 insertions(+), 20 deletions(-) create mode 100644 docs/iclr2021_conference.bst diff --git a/docs/db.tex b/docs/db.tex index 9f39cd6..50a5df1 100644 --- a/docs/db.tex +++ b/docs/db.tex @@ -190,22 +190,29 @@ \section{Experiments} \subsection{Noisy XOR} +todo: replace majority-layer with majority + +pathological: single bit-flip changes the classification + +following \cite{granmo18} + \begin{figure}[h] \centering \includegraphics[width=0.8\textwidth]{noisy-xor-architecture.png} - \caption{$\partial\mathbb{B}$ net for the noisy-xor problem.} + \caption{{\em A $\partial\mathbb{B}$ net for the noisy-xor problem}. The net concatenates the soft-bit input, ${\bf x}$ (length 12), with its negation, ${\bf 1 - x}$, and supplies the resulting vector (length 24) to a $\partial\text{AND-LAYER}$ (width 32), $\partial\text{OR-LAYER}$ (width 32), $\partial\text{NOT-LAYER}$ (width 16), and a final $\partial\text{MAJORITY}$ to produce a single soft-bit $y \in [0,1]$ (to predict odd parity) and its negation $1-y$ (to predict even parity). The net's weights, once hardened, consume $288$ bytes.} \label{fig:noisy-xor-architecture} \end{figure} + todo: weight initialization -\begin{table}[] +\begin{table}[h] \centering \begin{tabular}{llllll} \multicolumn{1}{c}{} & \multicolumn{5}{c}{accuracy} \\ \multicolumn{1}{l|}{} & \multicolumn{1}{l|}{mean} & \multicolumn{1}{l|}{5 \%ile} & \multicolumn{1}{l|}{95 \%ile} & \multicolumn{1}{l|}{min} & \multicolumn{1}{l|}{max} \\ \hline \multicolumn{1}{|l|}{Tsetlin} & \multicolumn{1}{l|}{99.3 +/- 0.3} & \multicolumn{1}{l|}{95.9} & \multicolumn{1}{l|}{100.0} & \multicolumn{1}{l|}{91.6} & \multicolumn{1}{l|}{100.0} \\ \hline - \multicolumn{1}{|l|}{$\partial\mathbb{B}$} & \multicolumn{1}{l|}{\textbf{97.9 +/- 0.2}} & \multicolumn{1}{l|}{\textbf{95.4}} & \multicolumn{1}{l|}{\textbf{100.0}} & \multicolumn{1}{l|}{\textbf{93.6}} & \multicolumn{1}{l|}{\textbf{100.0}} \\ \hline + \multicolumn{1}{|l|}{$\partial\mathbb{B}$ net} & \multicolumn{1}{l|}{\textbf{97.9 +/- 0.2}} & \multicolumn{1}{l|}{\textbf{95.4}} & \multicolumn{1}{l|}{\textbf{100.0}} & \multicolumn{1}{l|}{\textbf{93.6}} & \multicolumn{1}{l|}{\textbf{100.0}} \\ \hline \multicolumn{1}{|l|}{neural network} & \multicolumn{1}{l|}{95.4 +/- 0.5} & \multicolumn{1}{l|}{90.1} & \multicolumn{1}{l|}{98.6} & \multicolumn{1}{l|}{88.2} & \multicolumn{1}{l|}{99.9} \\ \hline \multicolumn{1}{|l|}{SVM} & \multicolumn{1}{l|}{58.0 +/- 0.3} & \multicolumn{1}{l|}{56.4} & \multicolumn{1}{l|}{59.2} & \multicolumn{1}{l|}{55.4} & \multicolumn{1}{l|}{66.5} \\ \hline \multicolumn{1}{|l|}{naive Bayes} & \multicolumn{1}{l|}{49.8 +/- 0.2} & \multicolumn{1}{l|}{48.3} & \multicolumn{1}{l|}{51.0} & \multicolumn{1}{l|}{41.3} & \multicolumn{1}{l|}{52.7} \\ \hline @@ -214,19 +221,6 @@ \subsection{Noisy XOR} \caption{Noisy-XOR results.} \label{tab:noisy-xor-results} \end{table} -\begin{comment} -""" -| Technique/Accuracy | Mean | 5 %ile | 95 %ile | Min | Max | -| ------------------- | -------------- | ------- | ------- | ------ | ------ | -| Tsetlin | 99.3 +/- 0.3 | 95.9 | 100.0 | 91.6 | 100.0 | -| dB | 97.9 +/- 0.2 | 95.4 | 100.0 | 93.6 | 100.0 | -| Neural network | 95.4 +/- 0.5 | 90.1 | 98.6 | 88.2 | 99.9 | -| SVM | 58.0 +/- 0.3 | 56.4 | 59.2 | 55.4 | 66.5 | -| Naive Bayes | 49.8 +/- 0.2 | 48.3 | 51.0 | 41.3 | 52.7 | -| Logistic regression | 49.8 +/- 0.3 | 47.8 | 51.1 | 41.1 | 53.1 | -Source: https://arxiv.org/pdf/1804.01508.pdf -""" -\end{comment} \section{Conclusion} @@ -236,8 +230,8 @@ \subsubsection*{Acknowledgments} acknowledgments, including those to funding agencies, go at the end of the paper. -\bibliography{iclr2021_conference} \bibliographystyle{iclr2021_conference} +\bibliography{db} \appendix diff --git a/docs/iclr2021_conference.bst b/docs/iclr2021_conference.bst new file mode 100644 index 0000000..149a48c --- /dev/null +++ b/docs/iclr2021_conference.bst @@ -0,0 +1,1440 @@ +%% File: `iclr2017.bst' +%% A copy of iclm2010.bst, which is a modification of `plainnl.bst' for use with natbib package +%% +%% Copyright 2010 Hal Daum\'e III +%% Modified by J. Frnkranz +%% - Changed labels from (X and Y, 2000) to (X & Y, 2000) +%% +%% Copyright 1993-2007 Patrick W Daly +%% Max-Planck-Institut f\"ur Sonnensystemforschung +%% Max-Planck-Str. 2 +%% D-37191 Katlenburg-Lindau +%% Germany +%% E-mail: daly@mps.mpg.de +%% +%% This program can be redistributed and/or modified under the terms +%% of the LaTeX Project Public License Distributed from CTAN +%% archives in directory macros/latex/base/lppl.txt; either +%% version 1 of the License, or any later version. +%% + % Version and source file information: + % \ProvidesFile{icml2010.mbs}[2007/11/26 1.93 (PWD)] + % + % BibTeX `plainnat' family + % version 0.99b for BibTeX versions 0.99a or later, + % for LaTeX versions 2.09 and 2e. + % + % For use with the `natbib.sty' package; emulates the corresponding + % member of the `plain' family, but with author-year citations. + % + % With version 6.0 of `natbib.sty', it may also be used for numerical + % citations, while retaining the commands \citeauthor, \citefullauthor, + % and \citeyear to print the corresponding information. + % + % For version 7.0 of `natbib.sty', the KEY field replaces missing + % authors/editors, and the date is left blank in \bibitem. + % + % Includes field EID for the sequence/citation number of electronic journals + % which is used instead of page numbers. + % + % Includes fields ISBN and ISSN. + % + % Includes field URL for Internet addresses. + % + % Includes field DOI for Digital Object Idenfifiers. + % + % Works best with the url.sty package of Donald Arseneau. + % + % Works with identical authors and year are further sorted by + % citation key, to preserve any natural sequence. + % +ENTRY + { address + author + booktitle + chapter + doi + eid + edition + editor + howpublished + institution + isbn + issn + journal + key + month + note + number + organization + pages + publisher + school + series + title + type + url + volume + year + } + {} + { label extra.label sort.label short.list } + +INTEGERS { output.state before.all mid.sentence after.sentence after.block } + +FUNCTION {init.state.consts} +{ #0 'before.all := + #1 'mid.sentence := + #2 'after.sentence := + #3 'after.block := +} + +STRINGS { s t } + +FUNCTION {output.nonnull} +{ 's := + output.state mid.sentence = + { ", " * write$ } + { output.state after.block = + { add.period$ write$ + newline$ + "\newblock " write$ + } + { output.state before.all = + 'write$ + { add.period$ " " * write$ } + if$ + } + if$ + mid.sentence 'output.state := + } + if$ + s +} + +FUNCTION {output} +{ duplicate$ empty$ + 'pop$ + 'output.nonnull + if$ +} + +FUNCTION {output.check} +{ 't := + duplicate$ empty$ + { pop$ "empty " t * " in " * cite$ * warning$ } + 'output.nonnull + if$ +} + +FUNCTION {fin.entry} +{ add.period$ + write$ + newline$ +} + +FUNCTION {new.block} +{ output.state before.all = + 'skip$ + { after.block 'output.state := } + if$ +} + +FUNCTION {new.sentence} +{ output.state after.block = + 'skip$ + { output.state before.all = + 'skip$ + { after.sentence 'output.state := } + if$ + } + if$ +} + +FUNCTION {not} +{ { #0 } + { #1 } + if$ +} + +FUNCTION {and} +{ 'skip$ + { pop$ #0 } + if$ +} + +FUNCTION {or} +{ { pop$ #1 } + 'skip$ + if$ +} + +FUNCTION {new.block.checka} +{ empty$ + 'skip$ + 'new.block + if$ +} + +FUNCTION {new.block.checkb} +{ empty$ + swap$ empty$ + and + 'skip$ + 'new.block + if$ +} + +FUNCTION {new.sentence.checka} +{ empty$ + 'skip$ + 'new.sentence + if$ +} + +FUNCTION {new.sentence.checkb} +{ empty$ + swap$ empty$ + and + 'skip$ + 'new.sentence + if$ +} + +FUNCTION {field.or.null} +{ duplicate$ empty$ + { pop$ "" } + 'skip$ + if$ +} + +FUNCTION {emphasize} +{ duplicate$ empty$ + { pop$ "" } + { "\emph{" swap$ * "}" * } + if$ +} + +INTEGERS { nameptr namesleft numnames } + +FUNCTION {format.names} +{ 's := + #1 'nameptr := + s num.names$ 'numnames := + numnames 'namesleft := + { namesleft #0 > } + { s nameptr "{ff~}{vv~}{ll}{, jj}" format.name$ 't := + nameptr #1 > + { namesleft #1 > + { ", " * t * } + { numnames #2 > + { "," * } + 'skip$ + if$ + t "others" = + { " et~al." * } + { " and " * t * } + if$ + } + if$ + } + 't + if$ + nameptr #1 + 'nameptr := + namesleft #1 - 'namesleft := + } + while$ +} + +FUNCTION {format.key} +{ empty$ + { key field.or.null } + { "" } + if$ +} + +FUNCTION {format.authors} +{ author empty$ + { "" } + { author format.names } + if$ +} + +FUNCTION {format.editors} +{ editor empty$ + { "" } + { editor format.names + editor num.names$ #1 > + { " (eds.)" * } + { " (ed.)" * } + if$ + } + if$ +} + +FUNCTION {format.isbn} +{ isbn empty$ + { "" } + { new.block "ISBN " isbn * } + if$ +} + +FUNCTION {format.issn} +{ issn empty$ + { "" } + { new.block "ISSN " issn * } + if$ +} + +FUNCTION {format.url} +{ url empty$ + { "" } + { new.block "URL \url{" url * "}" * } + if$ +} + +FUNCTION {format.doi} +{ doi empty$ + { "" } + { new.block "\doi{" doi * "}" * } + if$ +} + +FUNCTION {format.title} +{ title empty$ + { "" } + { title "t" change.case$ } + if$ +} + +FUNCTION {format.full.names} +{'s := + #1 'nameptr := + s num.names$ 'numnames := + numnames 'namesleft := + { namesleft #0 > } + { s nameptr + "{vv~}{ll}" format.name$ 't := + nameptr #1 > + { + namesleft #1 > + { ", " * t * } + { + numnames #2 > + { "," * } + 'skip$ + if$ + t "others" = + { " et~al." * } + { " and " * t * } + if$ + } + if$ + } + 't + if$ + nameptr #1 + 'nameptr := + namesleft #1 - 'namesleft := + } + while$ +} + +FUNCTION {author.editor.full} +{ author empty$ + { editor empty$ + { "" } + { editor format.full.names } + if$ + } + { author format.full.names } + if$ +} + +FUNCTION {author.full} +{ author empty$ + { "" } + { author format.full.names } + if$ +} + +FUNCTION {editor.full} +{ editor empty$ + { "" } + { editor format.full.names } + if$ +} + +FUNCTION {make.full.names} +{ type$ "book" = + type$ "inbook" = + or + 'author.editor.full + { type$ "proceedings" = + 'editor.full + 'author.full + if$ + } + if$ +} + +FUNCTION {output.bibitem} +{ newline$ + "\bibitem[" write$ + label write$ + ")" make.full.names duplicate$ short.list = + { pop$ } + { * } + if$ + "]{" * write$ + cite$ write$ + "}" write$ + newline$ + "" + before.all 'output.state := +} + +FUNCTION {n.dashify} +{ 't := + "" + { t empty$ not } + { t #1 #1 substring$ "-" = + { t #1 #2 substring$ "--" = not + { "--" * + t #2 global.max$ substring$ 't := + } + { { t #1 #1 substring$ "-" = } + { "-" * + t #2 global.max$ substring$ 't := + } + while$ + } + if$ + } + { t #1 #1 substring$ * + t #2 global.max$ substring$ 't := + } + if$ + } + while$ +} + +FUNCTION {format.date} +{ year duplicate$ empty$ + { "empty year in " cite$ * warning$ + pop$ "" } + 'skip$ + if$ + month empty$ + 'skip$ + { month + " " * swap$ * + } + if$ + extra.label * +} + +FUNCTION {format.btitle} +{ title emphasize +} + +FUNCTION {tie.or.space.connect} +{ duplicate$ text.length$ #3 < + { "~" } + { " " } + if$ + swap$ * * +} + +FUNCTION {either.or.check} +{ empty$ + 'pop$ + { "can't use both " swap$ * " fields in " * cite$ * warning$ } + if$ +} + +FUNCTION {format.bvolume} +{ volume empty$ + { "" } + { "volume" volume tie.or.space.connect + series empty$ + 'skip$ + { " of " * series emphasize * } + if$ + "volume and number" number either.or.check + } + if$ +} + +FUNCTION {format.number.series} +{ volume empty$ + { number empty$ + { series field.or.null } + { output.state mid.sentence = + { "number" } + { "Number" } + if$ + number tie.or.space.connect + series empty$ + { "there's a number but no series in " cite$ * warning$ } + { " in " * series * } + if$ + } + if$ + } + { "" } + if$ +} + +FUNCTION {format.edition} +{ edition empty$ + { "" } + { output.state mid.sentence = + { edition "l" change.case$ " edition" * } + { edition "t" change.case$ " edition" * } + if$ + } + if$ +} + +INTEGERS { multiresult } + +FUNCTION {multi.page.check} +{ 't := + #0 'multiresult := + { multiresult not + t empty$ not + and + } + { t #1 #1 substring$ + duplicate$ "-" = + swap$ duplicate$ "," = + swap$ "+" = + or or + { #1 'multiresult := } + { t #2 global.max$ substring$ 't := } + if$ + } + while$ + multiresult +} + +FUNCTION {format.pages} +{ pages empty$ + { "" } + { pages multi.page.check + { "pp.\ " pages n.dashify tie.or.space.connect } + { "pp.\ " pages tie.or.space.connect } + if$ + } + if$ +} + +FUNCTION {format.eid} +{ eid empty$ + { "" } + { "art." eid tie.or.space.connect } + if$ +} + +FUNCTION {format.vol.num.pages} +{ volume field.or.null + number empty$ + 'skip$ + { "\penalty0 (" number * ")" * * + volume empty$ + { "there's a number but no volume in " cite$ * warning$ } + 'skip$ + if$ + } + if$ + pages empty$ + 'skip$ + { duplicate$ empty$ + { pop$ format.pages } + { ":\penalty0 " * pages n.dashify * } + if$ + } + if$ +} + +FUNCTION {format.vol.num.eid} +{ volume field.or.null + number empty$ + 'skip$ + { "\penalty0 (" number * ")" * * + volume empty$ + { "there's a number but no volume in " cite$ * warning$ } + 'skip$ + if$ + } + if$ + eid empty$ + 'skip$ + { duplicate$ empty$ + { pop$ format.eid } + { ":\penalty0 " * eid * } + if$ + } + if$ +} + +FUNCTION {format.chapter.pages} +{ chapter empty$ + 'format.pages + { type empty$ + { "chapter" } + { type "l" change.case$ } + if$ + chapter tie.or.space.connect + pages empty$ + 'skip$ + { ", " * format.pages * } + if$ + } + if$ +} + +FUNCTION {format.in.ed.booktitle} +{ booktitle empty$ + { "" } + { editor empty$ + { "In " booktitle emphasize * } + { "In " format.editors * ", " * booktitle emphasize * } + if$ + } + if$ +} + +FUNCTION {empty.misc.check} +{ author empty$ title empty$ howpublished empty$ + month empty$ year empty$ note empty$ + and and and and and + key empty$ not and + { "all relevant fields are empty in " cite$ * warning$ } + 'skip$ + if$ +} + +FUNCTION {format.thesis.type} +{ type empty$ + 'skip$ + { pop$ + type "t" change.case$ + } + if$ +} + +FUNCTION {format.tr.number} +{ type empty$ + { "Technical Report" } + 'type + if$ + number empty$ + { "t" change.case$ } + { number tie.or.space.connect } + if$ +} + +FUNCTION {format.article.crossref} +{ key empty$ + { journal empty$ + { "need key or journal for " cite$ * " to crossref " * crossref * + warning$ + "" + } + { "In \emph{" journal * "}" * } + if$ + } + { "In " } + if$ + " \citet{" * crossref * "}" * +} + +FUNCTION {format.book.crossref} +{ volume empty$ + { "empty volume in " cite$ * "'s crossref of " * crossref * warning$ + "In " + } + { "Volume" volume tie.or.space.connect + " of " * + } + if$ + editor empty$ + editor field.or.null author field.or.null = + or + { key empty$ + { series empty$ + { "need editor, key, or series for " cite$ * " to crossref " * + crossref * warning$ + "" * + } + { "\emph{" * series * "}" * } + if$ + } + 'skip$ + if$ + } + 'skip$ + if$ + " \citet{" * crossref * "}" * +} + +FUNCTION {format.incoll.inproc.crossref} +{ editor empty$ + editor field.or.null author field.or.null = + or + { key empty$ + { booktitle empty$ + { "need editor, key, or booktitle for " cite$ * " to crossref " * + crossref * warning$ + "" + } + { "In \emph{" booktitle * "}" * } + if$ + } + { "In " } + if$ + } + { "In " } + if$ + " \citet{" * crossref * "}" * +} + +FUNCTION {article} +{ output.bibitem + format.authors "author" output.check + author format.key output + new.block + format.title "title" output.check + new.block + crossref missing$ + { journal emphasize "journal" output.check + eid empty$ + { format.vol.num.pages output } + { format.vol.num.eid output } + if$ + format.date "year" output.check + } + { format.article.crossref output.nonnull + eid empty$ + { format.pages output } + { format.eid output } + if$ + } + if$ + format.issn output + format.doi output + format.url output + new.block + note output + fin.entry +} + +FUNCTION {book} +{ output.bibitem + author empty$ + { format.editors "author and editor" output.check + editor format.key output + } + { format.authors output.nonnull + crossref missing$ + { "author and editor" editor either.or.check } + 'skip$ + if$ + } + if$ + new.block + format.btitle "title" output.check + crossref missing$ + { format.bvolume output + new.block + format.number.series output + new.sentence + publisher "publisher" output.check + address output + } + { new.block + format.book.crossref output.nonnull + } + if$ + format.edition output + format.date "year" output.check + format.isbn output + format.doi output + format.url output + new.block + note output + fin.entry +} + +FUNCTION {booklet} +{ output.bibitem + format.authors output + author format.key output + new.block + format.title "title" output.check + howpublished address new.block.checkb + howpublished output + address output + format.date output + format.isbn output + format.doi output + format.url output + new.block + note output + fin.entry +} + +FUNCTION {inbook} +{ output.bibitem + author empty$ + { format.editors "author and editor" output.check + editor format.key output + } + { format.authors output.nonnull + crossref missing$ + { "author and editor" editor either.or.check } + 'skip$ + if$ + } + if$ + new.block + format.btitle "title" output.check + crossref missing$ + { format.bvolume output + format.chapter.pages "chapter and pages" output.check + new.block + format.number.series output + new.sentence + publisher "publisher" output.check + address output + } + { format.chapter.pages "chapter and pages" output.check + new.block + format.book.crossref output.nonnull + } + if$ + format.edition output + format.date "year" output.check + format.isbn output + format.doi output + format.url output + new.block + note output + fin.entry +} + +FUNCTION {incollection} +{ output.bibitem + format.authors "author" output.check + author format.key output + new.block + format.title "title" output.check + new.block + crossref missing$ + { format.in.ed.booktitle "booktitle" output.check + format.bvolume output + format.number.series output + format.chapter.pages output + new.sentence + publisher "publisher" output.check + address output + format.edition output + format.date "year" output.check + } + { format.incoll.inproc.crossref output.nonnull + format.chapter.pages output + } + if$ + format.isbn output + format.doi output + format.url output + new.block + note output + fin.entry +} + +FUNCTION {inproceedings} +{ output.bibitem + format.authors "author" output.check + author format.key output + new.block + format.title "title" output.check + new.block + crossref missing$ + { format.in.ed.booktitle "booktitle" output.check + format.bvolume output + format.number.series output + format.pages output + address empty$ + { organization publisher new.sentence.checkb + organization output + publisher output + format.date "year" output.check + } + { address output.nonnull + format.date "year" output.check + new.sentence + organization output + publisher output + } + if$ + } + { format.incoll.inproc.crossref output.nonnull + format.pages output + } + if$ + format.isbn output + format.doi output + format.url output + new.block + note output + fin.entry +} + +FUNCTION {conference} { inproceedings } + +FUNCTION {manual} +{ output.bibitem + format.authors output + author format.key output + new.block + format.btitle "title" output.check + organization address new.block.checkb + organization output + address output + format.edition output + format.date output + format.url output + new.block + note output + fin.entry +} + +FUNCTION {mastersthesis} +{ output.bibitem + format.authors "author" output.check + author format.key output + new.block + format.title "title" output.check + new.block + "Master's thesis" format.thesis.type output.nonnull + school "school" output.check + address output + format.date "year" output.check + format.url output + new.block + note output + fin.entry +} + +FUNCTION {misc} +{ output.bibitem + format.authors output + author format.key output + title howpublished new.block.checkb + format.title output + howpublished new.block.checka + howpublished output + format.date output + format.issn output + format.url output + new.block + note output + fin.entry + empty.misc.check +} + +FUNCTION {phdthesis} +{ output.bibitem + format.authors "author" output.check + author format.key output + new.block + format.btitle "title" output.check + new.block + "PhD thesis" format.thesis.type output.nonnull + school "school" output.check + address output + format.date "year" output.check + format.url output + new.block + note output + fin.entry +} + +FUNCTION {proceedings} +{ output.bibitem + format.editors output + editor format.key output + new.block + format.btitle "title" output.check + format.bvolume output + format.number.series output + address output + format.date "year" output.check + new.sentence + organization output + publisher output + format.isbn output + format.doi output + format.url output + new.block + note output + fin.entry +} + +FUNCTION {techreport} +{ output.bibitem + format.authors "author" output.check + author format.key output + new.block + format.title "title" output.check + new.block + format.tr.number output.nonnull + institution "institution" output.check + address output + format.date "year" output.check + format.url output + new.block + note output + fin.entry +} + +FUNCTION {unpublished} +{ output.bibitem + format.authors "author" output.check + author format.key output + new.block + format.title "title" output.check + new.block + note "note" output.check + format.date output + format.url output + fin.entry +} + +FUNCTION {default.type} { misc } + + +MACRO {jan} {"January"} + +MACRO {feb} {"February"} + +MACRO {mar} {"March"} + +MACRO {apr} {"April"} + +MACRO {may} {"May"} + +MACRO {jun} {"June"} + +MACRO {jul} {"July"} + +MACRO {aug} {"August"} + +MACRO {sep} {"September"} + +MACRO {oct} {"October"} + +MACRO {nov} {"November"} + +MACRO {dec} {"December"} + + + +MACRO {acmcs} {"ACM Computing Surveys"} + +MACRO {acta} {"Acta Informatica"} + +MACRO {cacm} {"Communications of the ACM"} + +MACRO {ibmjrd} {"IBM Journal of Research and Development"} + +MACRO {ibmsj} {"IBM Systems Journal"} + +MACRO {ieeese} {"IEEE Transactions on Software Engineering"} + +MACRO {ieeetc} {"IEEE Transactions on Computers"} + +MACRO {ieeetcad} + {"IEEE Transactions on Computer-Aided Design of Integrated Circuits"} + +MACRO {ipl} {"Information Processing Letters"} + +MACRO {jacm} {"Journal of the ACM"} + +MACRO {jcss} {"Journal of Computer and System Sciences"} + +MACRO {scp} {"Science of Computer Programming"} + +MACRO {sicomp} {"SIAM Journal on Computing"} + +MACRO {tocs} {"ACM Transactions on Computer Systems"} + +MACRO {tods} {"ACM Transactions on Database Systems"} + +MACRO {tog} {"ACM Transactions on Graphics"} + +MACRO {toms} {"ACM Transactions on Mathematical Software"} + +MACRO {toois} {"ACM Transactions on Office Information Systems"} + +MACRO {toplas} {"ACM Transactions on Programming Languages and Systems"} + +MACRO {tcs} {"Theoretical Computer Science"} + + +READ + +FUNCTION {sortify} +{ purify$ + "l" change.case$ +} + +INTEGERS { len } + +FUNCTION {chop.word} +{ 's := + 'len := + s #1 len substring$ = + { s len #1 + global.max$ substring$ } + 's + if$ +} + +FUNCTION {format.lab.names} +{ 's := + s #1 "{vv~}{ll}" format.name$ + s num.names$ duplicate$ + #2 > + { pop$ " et~al." * } + { #2 < + 'skip$ + { s #2 "{ff }{vv }{ll}{ jj}" format.name$ "others" = + { " et~al." * } + { " \& " * s #2 "{vv~}{ll}" format.name$ * } + if$ + } + if$ + } + if$ +} + +FUNCTION {author.key.label} +{ author empty$ + { key empty$ + { cite$ #1 #3 substring$ } + 'key + if$ + } + { author format.lab.names } + if$ +} + +FUNCTION {author.editor.key.label} +{ author empty$ + { editor empty$ + { key empty$ + { cite$ #1 #3 substring$ } + 'key + if$ + } + { editor format.lab.names } + if$ + } + { author format.lab.names } + if$ +} + +FUNCTION {author.key.organization.label} +{ author empty$ + { key empty$ + { organization empty$ + { cite$ #1 #3 substring$ } + { "The " #4 organization chop.word #3 text.prefix$ } + if$ + } + 'key + if$ + } + { author format.lab.names } + if$ +} + +FUNCTION {editor.key.organization.label} +{ editor empty$ + { key empty$ + { organization empty$ + { cite$ #1 #3 substring$ } + { "The " #4 organization chop.word #3 text.prefix$ } + if$ + } + 'key + if$ + } + { editor format.lab.names } + if$ +} + +FUNCTION {calc.short.authors} +{ type$ "book" = + type$ "inbook" = + or + 'author.editor.key.label + { type$ "proceedings" = + 'editor.key.organization.label + { type$ "manual" = + 'author.key.organization.label + 'author.key.label + if$ + } + if$ + } + if$ + 'short.list := +} + +FUNCTION {calc.label} +{ calc.short.authors + short.list + "(" + * + year duplicate$ empty$ + short.list key field.or.null = or + { pop$ "" } + 'skip$ + if$ + * + 'label := +} + +FUNCTION {sort.format.names} +{ 's := + #1 'nameptr := + "" + s num.names$ 'numnames := + numnames 'namesleft := + { namesleft #0 > } + { + s nameptr "{vv{ } }{ll{ }}{ ff{ }}{ jj{ }}" format.name$ 't := + nameptr #1 > + { + " " * + namesleft #1 = t "others" = and + { "zzzzz" * } + { numnames #2 > nameptr #2 = and + { "zz" * year field.or.null * " " * } + 'skip$ + if$ + t sortify * + } + if$ + } + { t sortify * } + if$ + nameptr #1 + 'nameptr := + namesleft #1 - 'namesleft := + } + while$ +} + +FUNCTION {sort.format.title} +{ 't := + "A " #2 + "An " #3 + "The " #4 t chop.word + chop.word + chop.word + sortify + #1 global.max$ substring$ +} + +FUNCTION {author.sort} +{ author empty$ + { key empty$ + { "to sort, need author or key in " cite$ * warning$ + "" + } + { key sortify } + if$ + } + { author sort.format.names } + if$ +} + +FUNCTION {author.editor.sort} +{ author empty$ + { editor empty$ + { key empty$ + { "to sort, need author, editor, or key in " cite$ * warning$ + "" + } + { key sortify } + if$ + } + { editor sort.format.names } + if$ + } + { author sort.format.names } + if$ +} + +FUNCTION {author.organization.sort} +{ author empty$ + { organization empty$ + { key empty$ + { "to sort, need author, organization, or key in " cite$ * warning$ + "" + } + { key sortify } + if$ + } + { "The " #4 organization chop.word sortify } + if$ + } + { author sort.format.names } + if$ +} + +FUNCTION {editor.organization.sort} +{ editor empty$ + { organization empty$ + { key empty$ + { "to sort, need editor, organization, or key in " cite$ * warning$ + "" + } + { key sortify } + if$ + } + { "The " #4 organization chop.word sortify } + if$ + } + { editor sort.format.names } + if$ +} + + +FUNCTION {presort} +{ calc.label + label sortify + " " + * + type$ "book" = + type$ "inbook" = + or + 'author.editor.sort + { type$ "proceedings" = + 'editor.organization.sort + { type$ "manual" = + 'author.organization.sort + 'author.sort + if$ + } + if$ + } + if$ + " " + * + year field.or.null sortify + * + " " + * + cite$ + * + #1 entry.max$ substring$ + 'sort.label := + sort.label * + #1 entry.max$ substring$ + 'sort.key$ := +} + +ITERATE {presort} + +SORT + +STRINGS { longest.label last.label next.extra } + +INTEGERS { longest.label.width last.extra.num number.label } + +FUNCTION {initialize.longest.label} +{ "" 'longest.label := + #0 int.to.chr$ 'last.label := + "" 'next.extra := + #0 'longest.label.width := + #0 'last.extra.num := + #0 'number.label := +} + +FUNCTION {forward.pass} +{ last.label label = + { last.extra.num #1 + 'last.extra.num := + last.extra.num int.to.chr$ 'extra.label := + } + { "a" chr.to.int$ 'last.extra.num := + "" 'extra.label := + label 'last.label := + } + if$ + number.label #1 + 'number.label := +} + +FUNCTION {reverse.pass} +{ next.extra "b" = + { "a" 'extra.label := } + 'skip$ + if$ + extra.label 'next.extra := + extra.label + duplicate$ empty$ + 'skip$ + { "{\natexlab{" swap$ * "}}" * } + if$ + 'extra.label := + label extra.label * 'label := +} + +EXECUTE {initialize.longest.label} + +ITERATE {forward.pass} + +REVERSE {reverse.pass} + +FUNCTION {bib.sort.order} +{ sort.label 'sort.key$ := +} + +ITERATE {bib.sort.order} + +SORT + +FUNCTION {begin.bib} +{ preamble$ empty$ + 'skip$ + { preamble$ write$ newline$ } + if$ + "\begin{thebibliography}{" number.label int.to.str$ * "}" * + write$ newline$ + "\providecommand{\natexlab}[1]{#1}" + write$ newline$ + "\providecommand{\url}[1]{\texttt{#1}}" + write$ newline$ + "\expandafter\ifx\csname urlstyle\endcsname\relax" + write$ newline$ + " \providecommand{\doi}[1]{doi: #1}\else" + write$ newline$ + " \providecommand{\doi}{doi: \begingroup \urlstyle{rm}\Url}\fi" + write$ newline$ +} + +EXECUTE {begin.bib} + +EXECUTE {init.state.consts} + +ITERATE {call.type$} + +FUNCTION {end.bib} +{ newline$ + "\end{thebibliography}" write$ newline$ +} + +EXECUTE {end.bib} diff --git a/docs/iclr2021_conference.sty b/docs/iclr2021_conference.sty index 752784e..6606c34 100644 --- a/docs/iclr2021_conference.sty +++ b/docs/iclr2021_conference.sty @@ -85,15 +85,14 @@ {\LARGE\sc \@title\par} %\bottomtitlebar % \vskip 0.1in % minus \ificlrfinal -% \lhead{Published as a conference paper at ICLR 2021} - \lhead{draft} + \lhead{Published as a 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z-!sgt%^LoPP9_%_!t0#Sbvyjv(&PW+=S7}jer&nvoW#}TtC08X-gRInXZz7h{{tZ8 B?_&S} From 2aff2016d86c95af964dc8ae0716f83061facecc Mon Sep 17 00:00:00 2001 From: Ian Wright <55286208+Z80coder@users.noreply.github.com> Date: Fri, 10 Mar 2023 17:37:50 +0000 Subject: [PATCH 038/113] more writing; additional symbolic support --- neurallogic/hard_and.py | 2 +- neurallogic/hard_concatenate.py | 46 ++++++++++++++++++++++++++++++ neurallogic/hard_or.py | 2 +- neurallogic/hard_vmap.py | 24 ++++++++++++++++ neurallogic/symbolic_generation.py | 6 ++-- tests/test_noisy_xor.py | 28 ++++++++++-------- tests/test_symbolic_generation.py | 4 ++- 7 files changed, 94 insertions(+), 18 deletions(-) create mode 100644 neurallogic/hard_concatenate.py create mode 100644 neurallogic/hard_vmap.py diff --git a/neurallogic/hard_and.py b/neurallogic/hard_and.py index 794930e..e57113d 100644 --- a/neurallogic/hard_and.py +++ b/neurallogic/hard_and.py @@ -8,7 +8,7 @@ # TODO: seperate and operation from mask operation def soft_and_neuron(w, x): - x = jax.vmap(hard_masks.soft_mask_to_true, 0, 0)(w, x) + x = jax.vmap(hard_masks.soft_mask_to_true_margin, 0, 0)(w, x) return jax.numpy.min(x) diff --git a/neurallogic/hard_concatenate.py b/neurallogic/hard_concatenate.py new file mode 100644 index 0000000..2a79ae3 --- /dev/null +++ b/neurallogic/hard_concatenate.py @@ -0,0 +1,46 @@ +import jax +from flax import linen as nn + +from neurallogic import neural_logic_net, symbolic_generation + + +def soft_concatenate(x, axis): + return jax.numpy.concatenate(x, axis) + + +def hard_concatenate(x, axis): + return soft_concatenate(x, axis) + + +class SoftConcatenate(nn.Module): + axis: int + @nn.compact + def __call__(self, x): + return soft_concatenate(x, self.axis) + + +class HardConcatenate(nn.Module): + axis: int + @nn.compact + def __call__(self, x): + return hard_concatenate(x, self.axis) + + +class SymbolicConcatenate: + def __init__(self, axis): + self.hard_concatenate = HardConcatenate(axis) + + def __call__(self, x): + jaxpr = symbolic_generation.make_symbolic_flax_jaxpr( + self.hard_concatenate, x + ) + return symbolic_generation.symbolic_expression(jaxpr, x) + + +concatenate = neural_logic_net.select( + lambda x, axis: SoftConcatenate(axis)(x), + lambda x, axis: HardConcatenate(axis)(x), + lambda x, axis: SymbolicConcatenate(axis)(x), +) + +# TODO: add tests \ No newline at end of file diff --git a/neurallogic/hard_or.py b/neurallogic/hard_or.py index 1a8f822..417066a 100644 --- a/neurallogic/hard_or.py +++ b/neurallogic/hard_or.py @@ -13,7 +13,7 @@ # TODO: seperate out the or operation from the mask operation def soft_or_neuron(w, x): - x = jax.vmap(hard_masks.soft_mask_to_false, 0, 0)(w, x) + x = jax.vmap(hard_masks.soft_mask_to_false_margin, 0, 0)(w, x) return jax.numpy.max(x) diff --git a/neurallogic/hard_vmap.py b/neurallogic/hard_vmap.py new file mode 100644 index 0000000..5e164fe --- /dev/null +++ b/neurallogic/hard_vmap.py @@ -0,0 +1,24 @@ +import jax +import numpy + +from neurallogic import neural_logic_net + + +def soft_vmap(f): + return jax.vmap(f) + + +def hard_vmap(f): + return soft_vmap(f) + + +def symbolic_vmap(f): + return numpy.vectorize(f, otypes=[object]) + +vmap = neural_logic_net.select( + lambda f: soft_vmap(f[0]), + lambda f: hard_vmap(f[1]), + lambda f: symbolic_vmap(f[2]) +) + +# TODO: add tests diff --git a/neurallogic/symbolic_generation.py b/neurallogic/symbolic_generation.py index 734718b..9e974f2 100644 --- a/neurallogic/symbolic_generation.py +++ b/neurallogic/symbolic_generation.py @@ -14,9 +14,9 @@ def symbolic_bind(prim, *args, **params): - print('\nprimitive: ', prim.name) - print('\targs:\n\t\t', args) - print('\tparams\n\t\t: ', params) + #print('\nprimitive: ', prim.name) + #print('\targs:\n\t\t', args) + #print('\tparams\n\t\t: ', params) symbolic_outvals = { 'broadcast_in_dim': symbolic_primitives.symbolic_broadcast_in_dim, 'reshape': symbolic_primitives.symbolic_reshape, diff --git a/tests/test_noisy_xor.py b/tests/test_noisy_xor.py index 097b853..affd4a5 100644 --- a/tests/test_noisy_xor.py +++ b/tests/test_noisy_xor.py @@ -20,12 +20,14 @@ harden_layer, neural_logic_net, initialization, + symbolic_primitives, + hard_vmap, + hard_concatenate ) from tests import utils config.update("jax_enable_x64", True) - def check_symbolic(nets, data, trained_state, dropout_rng): x_training, y_training, x_test, y_test = data _, hard, symbolic = nets @@ -49,10 +51,14 @@ def check_symbolic(nets, data, trained_state, dropout_rng): ) print("hard_net: final test_accuracy: %.2f" % (hard_test_accuracy * 100)) assert numpy.isclose(test_accuracy, hard_test_accuracy, atol=0.0001) + if False: symbolic_weights = hard_weights # utils.make_symbolic(hard_weights) symbolic_trained_state = train_state.TrainState.create( - apply_fn=symbolic.apply, params=symbolic_weights, tx=optax.sgd(1.0, 1.0) + apply_fn=symbolic.apply, + params=symbolic_weights, + tx=optax.sgd(1.0, 1.0), + dropout_rng=dropout_rng, ) symbolic_input = hard_input.tolist() symbolic_test_accuracy = apply_hard_model( @@ -62,12 +68,11 @@ def check_symbolic(nets, data, trained_state, dropout_rng): "symbolic_net: final test_accuracy: %.2f" % (symbolic_test_accuracy * 100) ) assert numpy.isclose(test_accuracy, symbolic_test_accuracy, atol=0.0001) - if False: + if True: # CPU and GPU give different results, so we can't easily regress on a static symbolic expression symbolic_input = [f"x{i}" for i in range(len(hard_input[0].tolist()))] - symbolic_output = symbolic.apply({"params": symbolic_weights}, symbolic_input) - print("symbolic_output", symbolic_output[0][:10000]) - + # This simply checks that the symbolic output can be generated + symbolic_output = symbolic.apply({"params": hard_weights}, symbolic_input, training=False) num_features = 12 num_classes = 2 @@ -104,8 +109,8 @@ def get_data(): # N.B. We use marginal versions of and/or layers for this performance # mean: 97.89, sem: 0.15, min: 93.58, max: 100.00, 5%: 95.40, 95%: 100.00 def nln(type, x, training: bool): - y = jax.vmap(lambda x: 1 - x)(x) - x = jax.numpy.concatenate([x, y], axis=0) + y = hard_vmap.vmap(type)((lambda x: 1 - x, lambda x: 1 - x, lambda x: symbolic_primitives.symbolic_not(x)))(x) + x = hard_concatenate.concatenate(type)([x, y], 0) layer_size = 32 dtype = jax.numpy.float64 @@ -129,12 +134,11 @@ def nln(type, x, training: bool): x = x.reshape((1, layer_size * not_layer_size)) x = hard_majority.majority_layer(type)()(x) - z = jax.vmap(lambda x: 1 - x)(x) - x = jax.numpy.concatenate([x, z], axis=0) + z = hard_vmap.vmap(type)((lambda x: 1 - x, lambda x: 1 - x, lambda x: symbolic_primitives.symbolic_not(x)))(x) + x = hard_concatenate.concatenate(type)([x, z], 0) ######################################################## - x = harden_layer.harden_layer(type)(x) x = x.reshape((num_classes, int(x.shape[0] / num_classes))) x = x.sum(-1) return x @@ -277,7 +281,7 @@ def get_config(): config = ml_collections.ConfigDict() config.learning_rate = 0.01 config.batch_size = 5000 - config.num_epochs = 2000 + config.num_epochs = 2000 # 2000 for paper return config diff --git a/tests/test_symbolic_generation.py b/tests/test_symbolic_generation.py index 3fc8631..e715455 100644 --- a/tests/test_symbolic_generation.py +++ b/tests/test_symbolic_generation.py @@ -5,7 +5,7 @@ from neurallogic import (hard_and, hard_majority, hard_not, hard_or, hard_xor, harden, harden_layer, neural_logic_net, real_encoder, - symbolic_generation) + symbolic_generation, hard_concatenate, hard_vmap, symbolic_primitives) from tests import utils @@ -14,6 +14,8 @@ def nln(type, x, width): # lacks the correct tensor structure # x = real_encoder.real_encoder_layer(type)(2)(x) # x = x.ravel() + y = hard_vmap.vmap(type)((lambda x: 1 - x, lambda x: 1 - x, lambda x: symbolic_primitives.symbolic_not(x)))(x) + x = hard_concatenate.concatenate(type)([x, y], 0) x = hard_or.or_layer(type)(width)(x) x = hard_and.and_layer(type)(width)(x) x = hard_xor.xor_layer(type)(width)(x) From 1fbe580ce15d33fd374bf23cfe272e321f22112a Mon Sep 17 00:00:00 2001 From: Ian Wright Date: Mon, 13 Mar 2023 08:32:09 +0000 Subject: [PATCH 039/113] updated --- docs/db.tex | 33 +++++++++++++++++++------------- docs/iclr2021_conference.sty | 3 ++- docs/noisy-xor-architecture.png | Bin 32684 -> 32691 bytes 3 files changed, 22 insertions(+), 14 deletions(-) diff --git a/docs/db.tex b/docs/db.tex index 50a5df1..2aaf788 100644 --- a/docs/db.tex +++ b/docs/db.tex @@ -43,8 +43,8 @@ \maketitle \begin{abstract} - $\partial\mathbb{B}$ nets are real-valued, differentiable neural networks - trained by backpropagation that learn discrete boolean functions. + $\partial\mathbb{B}$ nets are differentiable neural networks + that learn discrete boolean functions by backpropagation. $\partial\mathbb{B}$ nets, once trained, `harden' to boolean functions with identical semantics and therefore, unlike existing approaches to neural network binarization, retain their accuracy. Experiments demonstrate that $\partial\mathbb{B}$ nets achieve competitive performance on standard machine learning problems yet are significantly more compact (due to 1-bit weights) and interpretable (due to the logical nature of the learnt function). \end{abstract} @@ -185,43 +185,50 @@ \subsection{Logical layers} \end{equation*} A single OR neuron maps $n$ soft-bit inputs to a single soft-bit value. A $\partial\text{OR-LAYER}$ of $n$ neurons maps $m$ soft-bit inputs to $n$ soft-bit outputs. +\subsection{Architectures} + +todo: architectures \section{Experiments} -\subsection{Noisy XOR} +\subsection{Binary Iris} -todo: replace majority-layer with majority +\subsection{Noisy XOR} -pathological: single bit-flip changes the classification +The noisy XOR dataset \citep{noisy-xor-dataset} is an adversarial parity problem with noisy non-informative features. The dataset consists of 10K examples with 12 boolean inputs and a target label (where 0 = odd and 1 = even) that is a XOR function of 2 inputs. The remaining 10 inputs are entirely random. We train on 50\% of the data where, additionally, 40\% of the labels are inverted. -following \cite{granmo18} - -\begin{figure}[h] +\begin{figure}[t] \centering \includegraphics[width=0.8\textwidth]{noisy-xor-architecture.png} \caption{{\em A $\partial\mathbb{B}$ net for the noisy-xor problem}. The net concatenates the soft-bit input, ${\bf x}$ (length 12), with its negation, ${\bf 1 - x}$, and supplies the resulting vector (length 24) to a $\partial\text{AND-LAYER}$ (width 32), $\partial\text{OR-LAYER}$ (width 32), $\partial\text{NOT-LAYER}$ (width 16), and a final $\partial\text{MAJORITY}$ to produce a single soft-bit $y \in [0,1]$ (to predict odd parity) and its negation $1-y$ (to predict even parity). The net's weights, once hardened, consume $288$ bytes.} \label{fig:noisy-xor-architecture} \end{figure} - -todo: weight initialization +We initialized the network described in figure \ref{fig:noisy-xor-architecture} using the XX policy and then trained for 2000 epochs with the RAdam optimizer \citep{Liu2020On} and softmax cross-entropy loss. We measure the accuracy of the final net on the test data (to avoid handpicking the best configuration). Table \ref{tab:noisy-xor-results} compares the $\partial\mathbb{B}$ net against other classifiers \citep{granmo18} and reports the mean accuracy with 95\% confidence intervals obtained over 100 replications of the experiment with different random seeds. \begin{table}[h] \centering \begin{tabular}{llllll} - \multicolumn{1}{c}{} & \multicolumn{5}{c}{accuracy} \\ + \cline{2-6} + \multicolumn{1}{c}{} & \multicolumn{5}{c}{\textbf{accuracy}} \\ \cline{2-6} \multicolumn{1}{l|}{} & \multicolumn{1}{l|}{mean} & \multicolumn{1}{l|}{5 \%ile} & \multicolumn{1}{l|}{95 \%ile} & \multicolumn{1}{l|}{min} & \multicolumn{1}{l|}{max} \\ \hline \multicolumn{1}{|l|}{Tsetlin} & \multicolumn{1}{l|}{99.3 +/- 0.3} & \multicolumn{1}{l|}{95.9} & \multicolumn{1}{l|}{100.0} & \multicolumn{1}{l|}{91.6} & \multicolumn{1}{l|}{100.0} \\ \hline - \multicolumn{1}{|l|}{$\partial\mathbb{B}$ net} & \multicolumn{1}{l|}{\textbf{97.9 +/- 0.2}} & \multicolumn{1}{l|}{\textbf{95.4}} & \multicolumn{1}{l|}{\textbf{100.0}} & \multicolumn{1}{l|}{\textbf{93.6}} & \multicolumn{1}{l|}{\textbf{100.0}} \\ \hline + \multicolumn{1}{|l|}{$\partial\mathbb{B}$} & \multicolumn{1}{l|}{\textbf{97.9 +/- 0.2}} & \multicolumn{1}{l|}{\textbf{95.4}} & \multicolumn{1}{l|}{\textbf{100.0}} & \multicolumn{1}{l|}{\textbf{93.6}} & \multicolumn{1}{l|}{\textbf{100.0}} \\ \hline \multicolumn{1}{|l|}{neural network} & \multicolumn{1}{l|}{95.4 +/- 0.5} & \multicolumn{1}{l|}{90.1} & \multicolumn{1}{l|}{98.6} & \multicolumn{1}{l|}{88.2} & \multicolumn{1}{l|}{99.9} \\ \hline \multicolumn{1}{|l|}{SVM} & \multicolumn{1}{l|}{58.0 +/- 0.3} & \multicolumn{1}{l|}{56.4} & \multicolumn{1}{l|}{59.2} & \multicolumn{1}{l|}{55.4} & \multicolumn{1}{l|}{66.5} \\ \hline \multicolumn{1}{|l|}{naive Bayes} & \multicolumn{1}{l|}{49.8 +/- 0.2} & \multicolumn{1}{l|}{48.3} & \multicolumn{1}{l|}{51.0} & \multicolumn{1}{l|}{41.3} & \multicolumn{1}{l|}{52.7} \\ \hline \multicolumn{1}{|l|}{logistic regression} & \multicolumn{1}{l|}{49.8 +/- 0.3} & \multicolumn{1}{l|}{47.8} & \multicolumn{1}{l|}{51.1} & \multicolumn{1}{l|}{41.1} & \multicolumn{1}{l|}{53.1} \\ \hline \end{tabular} - \caption{Noisy-XOR results.} + \caption{{\em Noisy-XOR results}} \label{tab:noisy-xor-results} \end{table} +The high noise causes logistic regression and naive Bayes to randomly guess. The SVM hardly performs better. In constrast, the multilayer neural network, Tsetlin machine \citep{granmo18}, and $\partial\mathbb{B}$ net all successfully learn the underlying XOR signal. The Tsetlin machine performs best on this problem, with the $\partial\mathbb{B}$ net second. + +todo: can Tsetlin machines be chained in differentiable architectures? + +\subsection{MNIST} + \section{Conclusion} diff --git a/docs/iclr2021_conference.sty b/docs/iclr2021_conference.sty index 6606c34..9eb0379 100644 --- a/docs/iclr2021_conference.sty +++ b/docs/iclr2021_conference.sty @@ -85,7 +85,8 @@ {\LARGE\sc \@title\par} %\bottomtitlebar % \vskip 0.1in % minus \ificlrfinal - \lhead{Published as a conference paper at ICLR 2021} + %\lhead{Published as a conference paper at ICLR 2021} + \lhead{draft} \def\And{\end{tabular}\hfil\linebreak[0]\hfil \begin{tabular}[t]{l}\bf\rule{\z@}{24pt}\ignorespaces}% \def\AND{\end{tabular}\hfil\linebreak[4]\hfil diff --git a/docs/noisy-xor-architecture.png b/docs/noisy-xor-architecture.png index a59bb70f43a755b48c2b71fbc9ae8a90d22cf019..1f4ec5eb36676282eaf4e9fcb74941a779c4b534 100644 GIT binary patch literal 32691 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z08HR1B56mQx%D=hmg4%>nTd6;eYVlzhBCLU~X1DZYqn3iD`#^=+Nr-WfNAzJzFSc^)+=lo@(P&L-m?v{@<4{!l4JQ z6xtMoiwgjN*DZ2&1-X#obFNWX=*pVFP#7A>0uZ&K94hJ0Z-7-aI|mdPH{?c^V)Iwj zCmL*!%)re8xMFpv3w` z+v1HcbxI5ARsdi1W&=stokm`d)cvb zk|@0jWoIX?oVOSd(_*djmUeZL8ck@+DECa+y+=GCQiqe=mQjdLEu`dlqb-Meh-P}1 s9i@Va(u2xb-m?O0gWdW6_~*x6uHSQ}YJn~C5P8Zm Date: Mon, 13 Mar 2023 11:22:13 +0000 Subject: [PATCH 040/113] v 1.0 of hard_count layer --- neurallogic/hard_count.py | 73 ++++++++++++++++++++++++++++++++++++ neurallogic/hard_majority.py | 4 -- neurallogic/hard_xor.py | 1 - tests/test_hard_count.py | 52 +++++++++++++++++++++++++ 4 files changed, 125 insertions(+), 5 deletions(-) create mode 100644 neurallogic/hard_count.py create mode 100644 tests/test_hard_count.py diff --git a/neurallogic/hard_count.py b/neurallogic/hard_count.py new file mode 100644 index 0000000..85573a6 --- /dev/null +++ b/neurallogic/hard_count.py @@ -0,0 +1,73 @@ +import jax +from flax import linen as nn + +from neurallogic import neural_logic_net, symbolic_generation + + +def high_to_low(x, y): + return jax.numpy.minimum(1 - x, y) + +def soft_count(x: jax.numpy.array) -> float: + """ + Returns an array of soft-bits, of length |x|+1, and where only 1 soft-bit is high. + The index of the high soft-bit indicates the total quantity of low and high bits in the input array. + i.e. if index i is high, then there are i low bits + + E.g. if x = [0.1, 0.9, 0.2, 0.6, 0.4], then the output is y=[low, low, low, high, low, low] + y[3] is high, which indicates that + - 3 bits are low + - 2 bits are high + + E.g. if x = [0.0, 0.2, 0.3, 0.1, 0.4], then the output is y=[low, low, low, low, low, high] + y[5] is high, which indicates that + - 5 bits are low + - 0 bits are high + + E.g. if x = [0.9, 0.8, 0.7, 0.6, 0.5], then the output is y=[high, low, low, low, low, low] + y[0] is high, which indicates that + - 0 bits are low + - 5 bits are high + """ + sorted_x = jax.numpy.sort(x, axis=-1) + low = jax.numpy.array([0.0]) + high = jax.numpy.array([1.0]) + sorted_x = jax.numpy.concatenate([low, sorted_x, high]) + return jax.vmap(high_to_low)(sorted_x[:-1], sorted_x[1:]) + +def hard_count(x: jax.numpy.array) -> bool: + return 1.0 + +soft_count_layer = jax.vmap(soft_count, in_axes=0) + +hard_count_layer = jax.vmap(hard_count, in_axes=0) + + +class SoftCountLayer(nn.Module): + @nn.compact + def __call__(self, x): + return soft_count_layer(x) + + +class HardCountLayer(nn.Module): + @nn.compact + def __call__(self, x): + return hard_count_layer(x) + + +class SymbolicCountLayer: + def __init__(self): + self.hard_count_layer = HardCountLayer() + + def __call__(self, x): + jaxpr = symbolic_generation.make_symbolic_flax_jaxpr( + self.hard_count_layer, x + ) + return symbolic_generation.symbolic_expression(jaxpr, x) + + +majority_layer = neural_logic_net.select( + lambda: SoftCountLayer(), + lambda: HardCountLayer(), + lambda: SymbolicCountLayer(), +) + diff --git a/neurallogic/hard_majority.py b/neurallogic/hard_majority.py index 8f208a0..941523a 100644 --- a/neurallogic/hard_majority.py +++ b/neurallogic/hard_majority.py @@ -79,7 +79,3 @@ def __call__(self, x): # where k is the number of high-soft bits required for a majority # and where k is a soft-bit parameter. Requires constructing # a piecewise-continuous function (as per notebook). - - -# TODO: construct a soft-count layer from sorting/majority approach -# output is 1 high-soft bit that indicates the number of high-soft bits in the input diff --git a/neurallogic/hard_xor.py b/neurallogic/hard_xor.py index b35e8ef..9c75815 100644 --- a/neurallogic/hard_xor.py +++ b/neurallogic/hard_xor.py @@ -14,7 +14,6 @@ def differentiable_xor(x, y): def soft_xor_neuron(w, x): # Conditionally include input bits, according to weights x = jax.vmap(hard_masks.soft_mask_to_false, 0, 0)(w, x) - x = jax.lax.reduce(x, jax.numpy.array(0, dtype=x.dtype), differentiable_xor, (0,)) return x diff --git a/tests/test_hard_count.py b/tests/test_hard_count.py new file mode 100644 index 0000000..b7f45ef --- /dev/null +++ b/tests/test_hard_count.py @@ -0,0 +1,52 @@ +import numpy +import jax + +from neurallogic import hard_count + +def test_soft_count(): + # 2 bits are high in a 7-bit input array, x + x = numpy.array([1.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0]) + y = hard_count.soft_count(x) + # We expect a 8-bit output array, y, where y[5] is the only high soft-bit (indicating that 5 soft-bits are low in the input) + expected_output = numpy.array([0.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0]) + print("soft_count", y) + assert numpy.allclose(y, expected_output) + + # Same example as above, except instead of 0s and 1s, we have soft-bits + x = numpy.array([0.9, 0.1, 0.1, 0.1, 0.1, 0.9, 0.1]) + y = hard_count.soft_count(x) + # We expect an 8-bit output array, y, where y[5] is the only high soft-bit (indicating that 5 soft-bits are low in the input) + expected_output = numpy.array([0.1, 0.1, 0.1, 0.1, 0.1, 0.9, 0.10000002, 0.10000002]) + print("soft_count", y) + assert numpy.allclose(y, expected_output) + + x = numpy.array([1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0]) + y = hard_count.soft_count(x) + # We expect an 8-bit output array, y, where no y[0] is high (indicating that 0 soft-bits are low in the input) + expected_output = numpy.array([1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]) + print("soft_count", y) + assert numpy.allclose(y, expected_output) + + # Same example as above, except instead of 0s and 1s, we have soft-bits + x = numpy.array([0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9]) + y = hard_count.soft_count(x) + # We expect an 8-bit output array, y, where y[0] is high (indicating that 0 soft-bits are low in the input) + expected_output = numpy.array([0.9, 0.10000002, 0.10000002, 0.10000002, 0.10000002, 0.10000002, 0.10000002, 0.10000002]) + print("soft_count", y) + assert numpy.allclose(y, expected_output) + + x = numpy.array([0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]) + y = hard_count.soft_count(x) + # We expect an 8-bit output array, y, where y[7] is the only high soft-bit (indicating that 7 soft-bits are low in the input) + expected_output = numpy.array([0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0]) + print("soft_count", y) + assert numpy.allclose(y, expected_output) + + # Same example as above, except instead of 0s and 1s, we have soft-bits + x = numpy.array([0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1]) + y = hard_count.soft_count(x) + # We expect an 7-bit output array, y, where y[7] is the only high soft-bit (indicating that 7 soft-bits are low in the input) + expected_output = numpy.array([0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.9]) + print("soft_count", y) + assert numpy.allclose(y, expected_output) + From 16b9f08bcb3cf7ea55dec5980ec2d7dcf27960ad Mon Sep 17 00:00:00 2001 From: Ian Wright <55286208+Z80coder@users.noreply.github.com> Date: Mon, 13 Mar 2023 17:12:31 +0000 Subject: [PATCH 041/113] experiment with hard count --- neurallogic/hard_count.py | 11 ++++--- tests/test_hard_count.py | 1 + tests/test_iris.py | 66 ++++++++++++++++++++++++++++++++++----- 3 files changed, 67 insertions(+), 11 deletions(-) diff --git a/neurallogic/hard_count.py b/neurallogic/hard_count.py index 85573a6..27e72d4 100644 --- a/neurallogic/hard_count.py +++ b/neurallogic/hard_count.py @@ -7,7 +7,7 @@ def high_to_low(x, y): return jax.numpy.minimum(1 - x, y) -def soft_count(x: jax.numpy.array) -> float: +def soft_count(x: jax.numpy.array): """ Returns an array of soft-bits, of length |x|+1, and where only 1 soft-bit is high. The index of the high soft-bit indicates the total quantity of low and high bits in the input array. @@ -34,8 +34,11 @@ def soft_count(x: jax.numpy.array) -> float: sorted_x = jax.numpy.concatenate([low, sorted_x, high]) return jax.vmap(high_to_low)(sorted_x[:-1], sorted_x[1:]) -def hard_count(x: jax.numpy.array) -> bool: - return 1.0 +def hard_count(x: jax.numpy.array): + # We simply count the number of low bits + num_low_bits = jax.numpy.sum(x <= 0.5, axis=-1) + return jax.nn.one_hot(num_low_bits, num_classes=x.shape[-1] + 1) + soft_count_layer = jax.vmap(soft_count, in_axes=0) @@ -65,7 +68,7 @@ def __call__(self, x): return symbolic_generation.symbolic_expression(jaxpr, x) -majority_layer = neural_logic_net.select( +count_layer = neural_logic_net.select( lambda: SoftCountLayer(), lambda: HardCountLayer(), lambda: SymbolicCountLayer(), diff --git a/tests/test_hard_count.py b/tests/test_hard_count.py index b7f45ef..0a5e15a 100644 --- a/tests/test_hard_count.py +++ b/tests/test_hard_count.py @@ -50,3 +50,4 @@ def test_soft_count(): print("soft_count", y) assert numpy.allclose(y, expected_output) +# TODO: test soft_count == hard_count \ No newline at end of file diff --git a/tests/test_iris.py b/tests/test_iris.py index 95f1508..6bad86f 100644 --- a/tests/test_iris.py +++ b/tests/test_iris.py @@ -6,6 +6,7 @@ import optax import scipy from flax.training import train_state +from jax.config import config from tqdm import tqdm from neurallogic import ( @@ -16,6 +17,7 @@ hard_not, hard_or, hard_xor, + hard_count, harden, harden_layer, neural_logic_net, @@ -24,6 +26,7 @@ ) from tests import utils +config.update("jax_enable_x64", True) def check_symbolic(nets, data, trained_state): x_training, y_training, x_test, y_test = data @@ -147,7 +150,7 @@ def nln_iris(type, x, training: bool): # mean: 94.18, sem: 0.13, min: 80.00, max: 100.00, 5%: 86.67, 95%: 100.00 # Using majority with margin # mean: 93.95, sem: 0.13, min: 76.67, max: 100.00, 5%: 86.67, 95%: 100.00 -def nln_binary_iris(type, x, training: bool): +def nln_binary_iris_1(type, x, training: bool): dtype = jax.numpy.float32 x = hard_masks.mask_to_true_layer(type)(120, dtype=dtype)(x) x = hard_majority.majority_layer(type)()(x) @@ -163,6 +166,47 @@ def nln_binary_iris(type, x, training: bool): x = x.sum(-1) return x +def nln_binary_iris(type, x, training: bool): + layer_size = 20 # 80 # 60 + dtype = jax.numpy.float64 + x = hard_masks.mask_to_true_layer(type)(layer_size, dtype=dtype)(x) + x = x.ravel() + """ + x = hard_and.and_layer(type)( + layer_size, + dtype=dtype, + weights_init=initialization.initialize_bernoulli(0.01, 0.3, 0.501), + )(x) + """ + """ + x = hard_or.or_layer(type)( + layer_size, + dtype=dtype, + weights_init=initialization.initialize_bernoulli(0.99, 0.499, 0.7), + )(x) + x = x.ravel() + """ + not_layer_size = 4 + x = hard_not.not_layer(type)( + not_layer_size, + dtype=dtype, + weights_init=initialization.initialize_uniform_range(0.499, 0.501), + )(x) + total_size = layer_size * num_features * not_layer_size + block_size = 40 + num_blocks = int(total_size / block_size) + x = x.reshape((num_blocks, block_size)) + x = hard_majority.majority_layer(type)()(x) + x = x.reshape((num_classes - 1, int(x.shape[0] / (num_classes - 1)))) + x = hard_majority.majority_layer(type)()(x) + x = jax.numpy.array([x]) # TODO: shouldn't need to do this + ######################################################## + x = hard_count.count_layer(type)()(x) + x = x.ravel() + x = x.reshape((num_classes, int(x.shape[0] / num_classes))) + x = x.sum(-1) + return x + def batch_nln_iris(type, x, training: bool): return jax.vmap(lambda x: nln_iris(type, x, training))(x) @@ -179,8 +223,8 @@ class TrainState(train_state.TrainState): def create_train_state(net, rng, dropout_rng, config): mock_input = jax.numpy.ones([1, num_features]) soft_weights = net.init(rng, mock_input, training=False)["params"] - tx = optax.sgd(config.learning_rate, config.momentum) - # tx = optax.radam(learning_rate=config.learning_rate) + #tx = optax.sgd(config.learning_rate, config.momentum) + tx = optax.radam(learning_rate=config.learning_rate) return TrainState.create( apply_fn=net.apply, params=soft_weights, tx=tx, dropout_rng=dropout_rng ) @@ -190,6 +234,12 @@ def create_train_state(net, rng, dropout_rng, config): def update_model(state, grads): return state.apply_gradients(grads=grads) +def my_loss(predictions, targets): + return jax.vmap(lambda x: jax.numpy.where(x < 0.45, 0.0, x*x))(predictions - targets) + +def my_loss(predictions, targets): + x = predictions - targets + return x*x def apply_model_with_grad_impl(state, features, labels, dropout_rng, training: bool): dropout_train_rng = jax.random.fold_in(key=dropout_rng, data=state.step) @@ -204,6 +254,8 @@ def loss_fn(params): one_hot = jax.nn.one_hot(labels, num_classes) loss = jax.numpy.mean( optax.softmax_cross_entropy(logits=logits, labels=one_hot) + #optax.l2_loss(logits, one_hot) + #my_loss(logits, one_hot) ) return loss, logits @@ -308,10 +360,10 @@ def apply_hard_model_to_data(state, features, labels): def get_config(): config = ml_collections.ConfigDict() - config.learning_rate = 0.01 + config.learning_rate = 0.001 # sgd = 0.1 config.momentum = 0.9 config.batch_size = 120 - config.num_epochs = 500 # 500 for paper + config.num_epochs = 4000 # 500 for paper return config @@ -327,7 +379,7 @@ def train_test_split(features, labels, rng, test_size=0.2): labels[test_idx], ) - +@pytest.mark.skip(reason="temporarily off") def test_iris(): # Train net if binary_iris: @@ -344,7 +396,7 @@ def test_iris(): rng = jax.random.PRNGKey(0) print(soft.tabulate(rng, features[0:1], training=False)) - num_experiments = 1 # 1000 for paper + num_experiments = 1000 # 1000 for paper final_test_accuracies = [] for i in range(num_experiments): # Split features and labels into 80% training and 20% test From 3fe441b51065f00fd2b7675adfa9418c34d816af Mon Sep 17 00:00:00 2001 From: Ian Wright <55286208+Z80coder@users.noreply.github.com> Date: Tue, 14 Mar 2023 11:53:27 +0000 Subject: [PATCH 042/113] simpler architecture with comparable performance --- tests/test_iris.py | 36 +++++++++--------------------------- 1 file changed, 9 insertions(+), 27 deletions(-) diff --git a/tests/test_iris.py b/tests/test_iris.py index 6bad86f..afcb7ab 100644 --- a/tests/test_iris.py +++ b/tests/test_iris.py @@ -23,6 +23,9 @@ neural_logic_net, real_encoder, initialization, + hard_vmap, + hard_concatenate, + symbolic_primitives ) from tests import utils @@ -167,47 +170,26 @@ def nln_binary_iris_1(type, x, training: bool): return x def nln_binary_iris(type, x, training: bool): - layer_size = 20 # 80 # 60 dtype = jax.numpy.float64 - x = hard_masks.mask_to_true_layer(type)(layer_size, dtype=dtype)(x) - x = x.ravel() - """ + y = hard_vmap.vmap(type)((lambda x: 1 - x, lambda x: 1 - x, lambda x: symbolic_primitives.symbolic_not(x)))(x) + x = hard_concatenate.concatenate(type)([x, y], 0) + layer_size = 16 x = hard_and.and_layer(type)( layer_size, dtype=dtype, weights_init=initialization.initialize_bernoulli(0.01, 0.3, 0.501), )(x) - """ - """ - x = hard_or.or_layer(type)( - layer_size, - dtype=dtype, - weights_init=initialization.initialize_bernoulli(0.99, 0.499, 0.7), - )(x) x = x.ravel() - """ - not_layer_size = 4 - x = hard_not.not_layer(type)( - not_layer_size, - dtype=dtype, - weights_init=initialization.initialize_uniform_range(0.499, 0.501), - )(x) - total_size = layer_size * num_features * not_layer_size - block_size = 40 - num_blocks = int(total_size / block_size) - x = x.reshape((num_blocks, block_size)) - x = hard_majority.majority_layer(type)()(x) x = x.reshape((num_classes - 1, int(x.shape[0] / (num_classes - 1)))) x = hard_majority.majority_layer(type)()(x) - x = jax.numpy.array([x]) # TODO: shouldn't need to do this ######################################################## + x = jax.numpy.array([x]) # TODO: shouldn't need to do this x = hard_count.count_layer(type)()(x) x = x.ravel() x = x.reshape((num_classes, int(x.shape[0] / num_classes))) x = x.sum(-1) return x - def batch_nln_iris(type, x, training: bool): return jax.vmap(lambda x: nln_iris(type, x, training))(x) @@ -360,10 +342,10 @@ def apply_hard_model_to_data(state, features, labels): def get_config(): config = ml_collections.ConfigDict() - config.learning_rate = 0.001 # sgd = 0.1 + config.learning_rate = 0.01 # sgd = 0.1 config.momentum = 0.9 config.batch_size = 120 - config.num_epochs = 4000 # 500 for paper + config.num_epochs = 4000 # 20000 # 500 for paper return config From ac8273a8f364d9b3872443b9541a2548d1e656c2 Mon Sep 17 00:00:00 2001 From: Ian Wright Date: Thu, 16 Mar 2023 15:47:39 +0000 Subject: [PATCH 043/113] more --- docs/db-net.png | Bin 0 -> 28719 bytes docs/db.tex | 166 +- docs/logic-gates.png | Bin 0 -> 238237 bytes docs/noisy-xor-architecture.png | Bin 32691 -> 34875 bytes docs/proofs.nb | 34902 +++++++++++++++++++++++++++++- 5 files 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zpHJ9pP0;+P9kpXo1|+XPhugF4s{h7R%OMY)!F-nEJUmOjX|~!Pf$cp1N8nHkq{2y! zd}7XBUa9zqWq)Z*7qKD+IWYT^{}6f1Ic_3s*tEV^XpnqCKT6dnDfu6RnUb9Fn8JVI zLtNK%?U4Bd_Ylz(SBL*^LaBcR+yAc+W>lt_nXw|o7wk>~242r+oQ^uZ8Ze-c$kWpv nI!Cw&K>_~X`UOJ|&ze^nnn;yAFl8leAAuN{-K^GgeDc2lR4zQ+ literal 0 HcmV?d00001 diff --git a/docs/db.tex b/docs/db.tex index 2aaf788..64da3ee 100644 --- a/docs/db.tex +++ b/docs/db.tex @@ -11,14 +11,13 @@ \usepackage{graphicx} \usepackage{comment} -\title{$\partial\mathbb{B}$ nets: learning boolean functions with\\backpropagation} +\title{$\partial\mathbb{B}$ nets learn boolean functions by\\backpropagation} % Authors must not appear in the submitted version. They should be hidden % as long as the \iclrfinalcopy macro remains commented out below. % Non-anonymous submissions will be rejected without review. -\author{Ian Wright \thanks{GitHub, Z80coder@github.com} -} +\author{Ian Wright} % The \author macro works with any number of authors. There are two commands % used to separate the names and addresses of multiple authors: \And and \AND. @@ -45,11 +44,19 @@ \begin{abstract} $\partial\mathbb{B}$ nets are differentiable neural networks that learn discrete boolean functions by backpropagation. - $\partial\mathbb{B}$ nets, once trained, `harden' to boolean functions with identical semantics and therefore, unlike existing approaches to neural network binarization, retain their accuracy. Experiments demonstrate that $\partial\mathbb{B}$ nets achieve competitive performance on standard machine learning problems yet are significantly more compact (due to 1-bit weights) and interpretable (due to the logical nature of the learnt function). + $\partial\mathbb{B}$ nets, once trained, `harden' to boolean functions with identical semantics. In consequence, the boolean functions have identical accuracy to the trained nets, unlike existing approaches to neural network binarization. Experiments demonstrate that $\partial\mathbb{B}$ nets achieve competitive performance on standard machine learning problems yet are significantly more compact (due to 1-bit weights) and interpretable (due to the logical nature of the learnt function). \end{abstract} \section{Introduction} +\begin{figure}[h] + \centering + \includegraphics[width=0.95\textwidth]{db-net.png} + \caption{{\em Learning discrete boolean-valued functions with a $\partial\mathbb{B}$ net.}} + \label{fig:noisy-xor-architecture} +\end{figure} + + \section{Related work} \section{$\partial\mathbb{B}$ nets} @@ -78,6 +85,41 @@ \subsection{Differentiable boolean functions} Weights are trainable soft-bits. A high weight implies the corresponding operation is masked out and therefore inactive. +TODO: requirements are smooth almost everywhere, strictly increasing or decreasing (i.e. no zero gradient). We also want to maximise fan-in information where necessary (for backprop error signal) but also minimize it where necessary (e.g. picking a representative bit we want to optimize). Representative bit when we want a vector-wise update; otherwise, min/max. + +TODO: min/max are a degenerate case of sorting/ordering + +Define +\begin{equation*} +\begin{aligned} +\operatorname{margin}: [0,1]^{n} \times 1,2,\dots,n &\to [0,1],\\ +({\bf x}, i) &\mapsto \bar{\bf x} \times \left|x_{i} - 1/2\right| \text{.} +\end{aligned} +\end{equation*} + +Define +\begin{equation*} +\begin{aligned} +\operatorname{representative-bit}: [0,1]^{n} \times 1,2,\dots,n &\to [0,1],\\ +({\bf x}, i) &\mapsto +\begin{cases} +1/2 + \operatorname{margin}({\bf x}, i) & \text{if } x_{i} > 1/2 \\ +x_{i} + \operatorname{margin}({\bf x}, i) & \text{otherwise,} +\end{cases} +\end{aligned} +\end{equation*} + +\begin{figure}[t] + \centering + \includegraphics[trim=0pt 0pt 0pt 0pt, clip, width=0.975\textwidth]{logic-gates.png} + \caption{{\em Gradient-rich versus gradient-sparse differentiable boolean functions.} Each column contains contour plots of functions $f(x,y)$ that are hard-equivalent to a boolean function (one of $\neg(x \oplus y)$, $x \wedge y$, $x \vee y$, or $x \Rightarrow y$). Every function is continuous and differentiable almost everywhere (white lines indicate non-continuous derivatives). The upper plots are gradient-sparse, where vertical and horizontal contours indicate the function is constant with respect to one of its inputs, i.e. $\partial f/\partial y = 0$ or $\partial f/\partial x = 0$. The lower plots are gradient-rich, where the curved contours indicate the function always varies with respect to any of its inputs, i.e. $\partial f/\partial y \neq 0$ and $\partial f/\partial x \neq 0$. $\partial \mathbb{B}$ nets use gradient-rich functions to ensure that error is always backpropagated to all inputs.} + \label{fig:and-plot} +\end{figure} + +\subsubsection{Not} + +TODO: this is not the not operator, but `learn not'. + Define \begin{equation*} \begin{aligned} @@ -88,23 +130,53 @@ \subsection{Differentiable boolean functions} where $w$ is a weight and $x$ is a soft-bit value. $\partial${NOT} is hard-equivalent to the boolean function $\neg(x \oplus w)$ (proposition \ref{prop:not}). If the weight is high then $\partial${NOT} is hard-equivalent to the boolean identity function; otherwise it is hard-equivalent to $\neg$. In consequence, we can learn to logically not, or simply pass through, the input value $x$. + +\subsubsection{And} + Define \begin{equation*} \begin{aligned} -\partial\text{AND}: [0,1]^{2} &\to [0,1], \\ - (x, y) &\mapsto +\partial\text{AND}: [0,1]^{n} &\to [0,1], \\ +{\bf x} &\mapsto \operatorname{representative-bit}({\bf x}, \operatorname{argmin}\limits_{i} x[i]) +\end{aligned} +\end{equation*} +$\partial${AND} is hard-equivalent to the boolean function $\bigwedge_{i=1}^{n} x_i$ (proposition \ref{prop:and}). + +\begin{equation*} +\partial\text{AND}(x, y) = \begin{cases} - 1/2 + 1/2(x + y)(m - 1/2) & \text{if } 2m > 1 \\ - m + 1/2(x + y)(1/2 - m) & \text{otherwise,} + 1/2 + 1/2(x + y)(\operatorname{min}(x,y) - 1/2) & \text{if } \operatorname{min}(x,y) > 1/2 \\ + \operatorname{min}(x,y) + 1/2(x + y)(1/2 - \operatorname{min}(x,y)) & \text{otherwise.} \end{cases} -\end{aligned} \end{equation*} -where $m=\min(x,y)$, and $x$ and $y$ are soft-bit values. + $\partial${AND} is hard-equivalent to the boolean function $x \wedge y$ (proposition \ref{prop:and}). + +\subsubsection{Or} + +Define +\begin{equation*} +\begin{aligned} +\partial\text{OR}: [0,1]^{n} &\to [0,1], \\ +{\bf x} &\mapsto \operatorname{representative-bit}({\bf x}, \operatorname{argmax}\limits_{i} x[i]) +\end{aligned} +\end{equation*} +$\partial${OR} is hard-equivalent to the boolean function $\bigvee_{i=1}^{n} x_i$ (proposition \ref{prop:or}). + Define \begin{equation*} \begin{aligned} +\partial\text{OR}(x, y) = +\begin{cases} +1/2 + 1/2(x + y)(\operatorname{max}(x,y) - 1/2) & \text{if } \operatorname{max}(x,y) > 1/2 \\ +\operatorname{max}(x,y) + 1/2(x + y)(1/2 - \operatorname{max}(x,y)) & \text{otherwise.} +\end{cases} +\end{aligned} +\end{equation*} +\begin{comment} +\begin{equation*} +\begin{aligned} \partial\text{OR}: [0,1]^{2} &\to [0,1], \\ (x, y) &\mapsto \begin{cases} @@ -113,9 +185,12 @@ \subsection{Differentiable boolean functions} \end{cases} \end{aligned} \end{equation*} -where $m=\max(x,y)$, and $x$ and $y$ are soft-bit values. +\end{comment} + $\partial${OR} is hard-equivalent to the boolean function $x \vee y$ (proposition \ref{prop:or}). +\subsubsection{Implies} + Define \begin{equation*} \begin{aligned} @@ -126,17 +201,25 @@ \subsection{Differentiable boolean functions} where $w$ is a weight and $x$ is a soft-bit value. $\partial${IMPLIES} is hard-equivalent to the boolean function $w \Rightarrow x$ (proposition \ref{prop:implies}). +\subsubsection{Majority} + Define $\operatorname{majority-index}: \mathbb{Z}_{>0} \to \mathbb{Z}_{> 0}$ as $n \mapsto 1 + \lfloor \frac{n-1}{2}\rfloor$. -Define $\operatorname{select}: [0,1]^n \times {1, 2, \ldots, n} \to [0,1]$ as $({\bf x}, i) \mapsto x_{i}$. +%Define $\operatorname{select}: [0,1]^n \times {1, 2, \ldots, n} \to [0,1]$ as $({\bf x}, i) \mapsto x_{i}$. -Define $\operatorname{majority-bit}: [0,1]^n \to [0,1]$ as ${\bf x} \mapsto \operatorname{select}( \operatorname{sort}({\bf x}), \operatorname{majority-index}(\lvert{\bf x}\rvert))$, where $\operatorname{sort}({\bf x})$ sorts the elements of ${\bf x}$ in ascending order. +%Define $\operatorname{majority-bit}: [0,1]^n \to [0,1]$ as ${\bf x} \mapsto \operatorname{select}( \operatorname{sort}({\bf x}), \operatorname{majority-index}(\lvert{\bf x}\rvert))$, where $\operatorname{sort}({\bf x})$ sorts the elements of ${\bf x}$ in ascending order. -Define $\operatorname{majority-delta}: [0,1]^n \to [0,1]$ as -${\bf x} \mapsto \bar{\bf x} \times \left| \operatorname{majority-bit}({\bf x}) - 1/2\right|$. +%Define $\operatorname{majority-delta}: [0,1]^n \to [0,1]$ as ${\bf x} \mapsto \operatorname{margin}(\operatorname{sort}({\bf x}), \operatorname{majority-index}({\bf x}))$. Define \begin{equation*} +\begin{aligned} + \partial\text{MAJORITY}: [0,1]^{n} &\to [0,1], \\ + {\bf x} &\mapsto \operatorname{representative-bit}(\operatorname{sort}({\bf x}), \operatorname{majority-index}({\bf x})) +\end{aligned} +\end{equation*} +\begin{comment} +\begin{equation*} \begin{aligned} \partial\text{MAJORITY}: [0,1]^{n} &\to [0,1], \\ {\bf x} &\mapsto @@ -146,9 +229,49 @@ \subsection{Differentiable boolean functions} \end{cases} \end{aligned} \end{equation*} -where ${\bf x}$ is a vector of soft-bits, $m = \operatorname{majority-bit}({\bf x})$ and $\delta = \operatorname{majority-delta}({\bf x})$. +\end{comment} +where ${\bf x}$ is a vector of soft-bits, and $\operatorname{sort}({\bf x})$ sorts the elements of ${\bf x}$ in ascending order. $\partial${MAJORITY} is hard-equivalent to the boolean majority function (proposition \ref{prop:majority}). +TODO: explain how we use sort to avoid space blow-up. +TODO: discuss time-complexity of sort. + +\subsubsection{Count} + +Define +\begin{equation*} +\begin{aligned} +\operatorname{low-high}: [0,1]^{n} &\to [0,1]^{n+1},\\ +{\bf x} &\mapsto \left[ \operatorname{min}(1, x_{1}), \operatorname{min}(1 - x_{1}, x_{2}), \dots, \operatorname{min}(1 - x_{n-1}, x_{n}), \operatorname{min}(x_{n}, 1) \right]\text{.} +\end{aligned} +\end{equation*} + +Define +\begin{equation*} +\begin{aligned} +\partial\text{COUNT}: [0,1]^{n} &\to [0,1]^{n+1}, \\ +{\bf x} &\mapsto \operatorname{low-high}(\operatorname{sort}({\bf x}))\text{,} +\end{aligned} +\end{equation*} +where ${\bf x}$ is a vector of soft-bits. $\partial${COUNT} is hard-equivalent to +\begin{equation*} +\begin{aligned} +\text{COUNT}: \{0,1\}^{n} &\to \{0,1\}^{n+1}, \\ +{\bf x} &\mapsto \left[\operatorname{k-of-n}({\bf x}, 0), \operatorname{k-of-n}({\bf x}, 1), \dots, \operatorname{k-of-n}({\bf x}, n)\right]\text{,} +\end{aligned} +\end{equation*} +where +\begin{equation*} +\operatorname{k-of-n}({\bf x}, k) = \bigvee_{|S|=k} \bigwedge_{i\in S} x_i \bigwedge_{j\notin S} \neg x_j +\end{equation*} +(see proposition \ref{prop:count}). + +%$f(x_1, x_2, \ldots, x_n) = \bigvee_{|S|=k} \left( \bigwedge_{i\in S} x_i \bigwedge_{j\notin S} \neg x_j \right)$ +%is a logical OR that is taken over all possible subsets $S$ of size $k$ of the set ${1,2,\ldots,n}$. + +TODO: count is 1-hot function +TODO: count allows us to implement boolean counting functions. + \subsection{Logical layers} Define @@ -189,6 +312,8 @@ \subsection{Architectures} todo: architectures +todo: multi-label classification + \section{Experiments} \subsection{Binary Iris} @@ -219,7 +344,7 @@ \subsection{Noisy XOR} \multicolumn{1}{|l|}{naive Bayes} & \multicolumn{1}{l|}{49.8 +/- 0.2} & \multicolumn{1}{l|}{48.3} & \multicolumn{1}{l|}{51.0} & \multicolumn{1}{l|}{41.3} & \multicolumn{1}{l|}{52.7} \\ \hline \multicolumn{1}{|l|}{logistic regression} & \multicolumn{1}{l|}{49.8 +/- 0.3} & \multicolumn{1}{l|}{47.8} & \multicolumn{1}{l|}{51.1} & \multicolumn{1}{l|}{41.1} & \multicolumn{1}{l|}{53.1} \\ \hline \end{tabular} - \caption{{\em Noisy-XOR results}} + \caption{{\em Ranked noisy-XOR results}} \label{tab:noisy-xor-results} \end{table} @@ -353,6 +478,13 @@ \section{Proofs} \end{proof} \end{prop} +\begin{prop}\label{prop:count} + $\partial${COUNT} is hard-equivalent to COUNT. + \begin{proof} + todo + \end{proof} +\end{prop} + \end{document} diff --git a/docs/logic-gates.png b/docs/logic-gates.png new file mode 100644 index 0000000000000000000000000000000000000000..fffe6afc370bce4ef05176f925180d8b180b867b GIT binary patch literal 238237 zcmce;c|2C_7e2a?5)B$2LP(K$D54CJLQ#^LOcjMt$vjt-3`rSMLdsCaOqnT^DP>B= zqJ*L{&zx)be&6r=et)0O`R9DjIs5Yt^*qns``-7x?zOIUt!wR@8tN)*=~?Lsf>?Xx z@F7itpu0>EE8^&=@yXIEsF5^WAN*y-iErb-PS~wJC7}D*)tR@v(2<2P zn2AYz@-YY9Vm@=3S;IQ-@@Jo}zpE-XUi@9v+BuLXHkrzx1uoPds{&OB@ocLv9yY)Ov)>^9r3&dE zwZMn64odnCS{KebIG?h&AP(AFoN}-*-{x%PV7bk~(qfz51)H<>r);+!J!N;!@zgns zZPMGEot;IT|N2vg!Gq!!!Q0Ai-HVHt_B(scA8WUfY_C7- zx_?PJ>fzX}NOyZD6Y&Ad3AYjq?ytjVb&fUj-+v)AEIy_f6WS3`Q3e_sajn+I#>TDb 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"]"}]& )], + TemplateBox[{ + GraphicsBox[{ + GraphicsComplexBox[CompressedData[" +1:eJyMfQV4VEfUNlIo7u7a4g7FD+5SvFBcipTiTnF3K+7BCRogSJAYgRAkrhDZ +jSeboMXh39x735l/zna/7+PJ8wxn7tzR4+fM3Yqjp/YdlyVTpkwNCmXKlFFu +r9LV+udEmYx/Ouwu4Ajnv6x/Puy5v4C1xxHBAg7XKwS8TWtvEnDlDHB7rICv +ac3jBdxZGyBJwGFafykC/lObkEXA+r+XDJbrwnoAYz3qc38BYz2AsR7AWA9g +rAcw1gMY6wGM9QDGetRzeMlgeS58Xep63Nl63Nl63Nl63Nl63Nl63Nl63Nl6 +3Nl63Nl63Nl6bPHqv8+Fr8tfwFgPYKwHMNYDGOsBjPUAxnoAYz2AsR6VDl4y +WNLFf+OVu806AKvn48/Ox5+djz87H392Pv7sfPzZ+fiz8/Fn52NL1/9NF+42 +58DXARjrAYz1AMZ6AGM9gLEewFgPYKxH5UMvGSz50n/TtbsNHvFz4OsAjPUA +xnoAYz2AsR7AWA9grEfloy9txvv2PeNflBgPMMYDjPEAYzzAGA8wxlNhycf/ +mw+629Adx1t+7nzf+LoAY32AsT7AWB9grA8w1qfKoZdkTy5hfZwPcj7C6ZDj +MccDfk58XYCxPsBYH2CsDzDWp8rVl2RPzmJ9nK9zvsj5CqdLjtcc7/g58XUB +xvoAY32AsT5VT3hJ9vQGrI/LKc7nOZ/kfIbTKacjjnf8nPi6AGN9gLE+Ve95 +Sfb0IKyPy10utzjf53yT8x3OFzgdcbzj58TXBRjrU/W4l2RPr8P6uB7B5TCX +Y1wOcD7K+RznC5yOON7xc+LrUvXSl2RPT8X6/lsv4noFl8tcrr20kROcj3O+ 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"}"}]}], "]"}]& )], + TemplateBox[{ + GraphicsBox[{ + GraphicsComplexBox[CompressedData[" +1:eJx9XQd4VcXyBxsootgQxN59imBFRVh9FnwKKorwsKHos/LEiqAiPGyIBQE7 +RQyiIIoaJJQghAuEhFCSmxASkkBubpKbHkSwi/975+xvxplr/nx+3/F395zd +ndnZaTvn5IShw2/6z16tWrU69qBWrRJX+2/yyf+K/5fqNA4xLkn7b/y/HNMe +ZkzNJYWMtwY/MN7zZ+LfdsaT6PkKxicl4ORKxgvp8WrGfWjAWsbF1H8944dp +go2MLV1/fw0xBn26PcwY9AGDPmDQBwz6gEEfMOgDBn3AoE+vyw6DZZ3+ni65 +avpChr6QoS9k6AsZ+kKGvpChL2ToCxn6Qg7rpeVsh8Eid3+/TkKXvYI+YNAH +DPqAQR8w6AMGfcCgDxjrBQz69L7ZYbDso7+Xu1ASHbpdrnr9wmb9wmb9wmb9 +wmb9wma9wkYew2b9wmb9kvXC3++jUNK6WDqA7RX0AYM+YNAHjPUCBn3AoA8Y +9Gm9tsNg0XN/rxdCSXJm18XSAWyvoA8Y6wUM+oBBHzDoAwZ9Wk/vcC3p7b/X +c6GkfWPlzK6LpQMY6wUM+oBBHzDoAwZ9wKBP250driU79Pd6O5SkB+y+sXJm +6QC2V9AHDPqAQR8w6AMGfdqO7nAt2VXQZ/W01WtWD1g5s+ti6QC2V9AHDPqA +QR8w6NN+wQ7Xkp8A+qzdsXra6gG7b6yc2XWxdADbK+gDBn3AoE/7OTtcS34P +6LN21Oppq9esHrD7xsqZXRdLB7C9gj5g0Kf9th2uJT8O9Fk7au2O1dNWr1k9 +YPeNlTO7LpYOYHsFfdoP3eFa8kuxXtYvsHbU2h2rp61es3rA7hsrZ3ZdLB3A 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b/docs/db.tex @@ -49,15 +49,25 @@ \section{Introduction} +Neural networks are typically trained by gradient descent in weight-space, where the direction of descent minimises some loss function. The backpropagation algorithm \citep{rumelhart1986learning} is a highly efficient method to calculate the partial derivatives of the loss with respect to the network's weights. In consequence, typical neural networks are differentiable functions with weights represented by machine floats. This approach has led to tremendous advances in machine learning. + +However, there are drawbacks. First, we cannot easily learn discrete functions, such as logical predicates. In consequence, interpreting or verifying what a network has learned is difficult. Also, many learning problems naturally arise in boolean rather than real-valued domains. Second, representing weights as machine floats confers the benefit of time-efficient training but at the cost of memory-inefficient trained models. For example, network quantisation techniques (TODO:refs) demonstrate that full 64 of 32-bit precision weights are often unnecessary for a network's predictive performance, although there is a trade-off. + +This paper proposes a new approach to mitigating these drawbacks. The main idea is to define a type of neural network, called a $\partial \mathbb{B}$ net, which has two aspects: a `soft' aspect, which is a differentiable real-valued function trainable by backpropagation, and a `hard' aspect, which is a non-differentiable boolean function. Both aspects are semantically equivalent. We train the `soft' net as normal, using backpropagation, then `harden' the network and its weights to yield a learned boolean function with equivalent predictive performance (see figure \ref{fig:main-idea}). In consequence, interpreting and verifying what a $\partial \mathbb{B}$ net has learned is relatively less difficult. The option to bias towards learning boolean functions will reduce variance in some domains. And boolean-valued, 1-bit weights increase the memory-efficiency of trained models. + +The main contributions of this work are (i) defining novel activation functions that harden to semantically equivalent boolean functions, (ii) defining novel network architectures that effectively learn boolean functions to solve multi-class classification problems, and (iii) experiments that demonstrate $\partial \mathbb{B}$ nets compete with existing approaches in terms of predictive performance yet yield considerably smaller models. + +This paper examines related work (section \ref{sec:related-work}), (ii) etc. + \begin{figure}[h] \centering - \includegraphics[width=0.95\textwidth]{db-net.png} - \caption{{\em Learning discrete boolean-valued functions with a $\partial\mathbb{B}$ net.}} - \label{fig:noisy-xor-architecture} + \includegraphics[width=0.9\textwidth]{db-net.png} + \caption{{\em Learning discrete boolean-valued functions with a $\partial\mathbb{B}$ net.} A $\partial \mathbb{B}$ net specifies (i) a differentiable neural network that is hard-equivalent to (ii) a non-differentiable discrete function. The neural network is trained as normal with backpropagation to yield a set of real weights. The real weights are hardened to boolean values and then bound with the discrete function. The result is a learned discrete function that performs identically to the trained network.} + \label{fig:main-idea} \end{figure} -\section{Related work} +\section{Related work}\label{sec:related-work} \section{$\partial\mathbb{B}$ nets} @@ -66,7 +76,7 @@ \section{$\partial\mathbb{B}$ nets} \end{definition} \begin{definition}[Hardening] -The {\em hardening} function, $h(x_{1}, \dots, x_{n}) = [f(x_{1}), \dots, f(x_{n})]$, converts soft-bits to hard-bits, where +The {\em hardening} function, $\operatorname{harden}(x_{1}, \dots, x_{n}) = [f(x_{1}), \dots, f(x_{n})]$, converts soft-bits to hard-bits, where \begin{equation*} f(x) = \begin{cases} @@ -78,7 +88,10 @@ \section{$\partial\mathbb{B}$ nets} \begin{definition}[Hard-equivalence] A function, $f: [0,1]^n \rightarrow [0,1]^m$, is {\em hard-equivalent} to a boolean function, $g: \{1,0\}^n \rightarrow \{1,0\}^m$, if - $h(f(h({\bf x}))) = g(h({\bf x}))$ for all ${\bf x} \in [0,1]^{n}$. + \begin{equation*} + \operatorname{harden}(f(\operatorname{harden}({\bf x}))) = g(\operatorname{harden}({\bf x})) + \end{equation*} +for all ${\bf x} \in [0,1]^{n}$. \end{definition} \subsection{Differentiable boolean functions} @@ -100,7 +113,7 @@ \subsection{Differentiable boolean functions} Define \begin{equation*} \begin{aligned} -\operatorname{representative-bit}: [0,1]^{n} \times 1,2,\dots,n &\to [0,1],\\ +\operatorname{augmented-bit}: [0,1]^{n} \times 1,2,\dots,n &\to [0,1],\\ ({\bf x}, i) &\mapsto \begin{cases} 1/2 + \operatorname{margin}({\bf x}, i) & \text{if } x_{i} > 1/2 \\ @@ -137,7 +150,7 @@ \subsubsection{And} \begin{equation*} \begin{aligned} \partial\text{AND}: [0,1]^{n} &\to [0,1], \\ -{\bf x} &\mapsto \operatorname{representative-bit}({\bf x}, \operatorname{argmin}\limits_{i} x[i]) +{\bf x} &\mapsto \operatorname{augmented-bit}({\bf x}, \operatorname{argmin}\limits_{i} x[i]) \end{aligned} \end{equation*} $\partial${AND} is hard-equivalent to the boolean function $\bigwedge_{i=1}^{n} x_i$ (proposition \ref{prop:and}). @@ -159,7 +172,7 @@ \subsubsection{Or} \begin{equation*} \begin{aligned} \partial\text{OR}: [0,1]^{n} &\to [0,1], \\ -{\bf x} &\mapsto \operatorname{representative-bit}({\bf x}, \operatorname{argmax}\limits_{i} x[i]) +{\bf x} &\mapsto \operatorname{augmented-bit}({\bf x}, \operatorname{argmax}\limits_{i} x[i]) \end{aligned} \end{equation*} $\partial${OR} is hard-equivalent to the boolean function $\bigvee_{i=1}^{n} x_i$ (proposition \ref{prop:or}). @@ -215,7 +228,7 @@ \subsubsection{Majority} \begin{equation*} \begin{aligned} \partial\text{MAJORITY}: [0,1]^{n} &\to [0,1], \\ - {\bf x} &\mapsto \operatorname{representative-bit}(\operatorname{sort}({\bf x}), \operatorname{majority-index}({\bf x})) + {\bf x} &\mapsto \operatorname{augmented-bit}(\operatorname{sort}({\bf x}), \operatorname{majority-index}({\bf x})) \end{aligned} \end{equation*} \begin{comment} @@ -314,6 +327,16 @@ \subsection{Architectures} todo: multi-label classification +Define +\begin{equation*} +\begin{aligned} +\partial\text{HARDEN}: [0,1]^{n} &\to [0,1]^{n}, \\ +{\bf x} &\mapsto \operatorname{harden}({\bf x})\text{.} +\end{aligned} +\end{equation*} +A $\partial\text{HARDEN}$ layer maps $m$ soft-bit inputs to $n$ hard-bit outputs. 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f83190c9cd06422527583412287c310dd6cfe316 Mon Sep 17 00:00:00 2001 From: Ian Wright Date: Fri, 17 Mar 2023 16:55:30 +0000 Subject: [PATCH 045/113] more --- docs/db.tex | 11 +- docs/logic-gates.png | Bin 238237 -> 251887 bytes docs/proofs.nb | 40300 +++++++++++++++++++++++++++++++++++++---- 3 files changed, 36732 insertions(+), 3579 deletions(-) diff --git a/docs/db.tex b/docs/db.tex index 89c961c..d9c1d53 100644 --- a/docs/db.tex +++ b/docs/db.tex @@ -55,7 +55,7 @@ \section{Introduction} This paper proposes a new approach to mitigating these drawbacks. The main idea is to define a type of neural network, called a $\partial \mathbb{B}$ net, which has two aspects: a `soft' aspect, which is a differentiable real-valued function trainable by backpropagation, and a `hard' aspect, which is a non-differentiable boolean function. Both aspects are semantically equivalent. We train the `soft' net as normal, using backpropagation, then `harden' the network and its weights to yield a learned boolean function with equivalent predictive performance (see figure \ref{fig:main-idea}). In consequence, interpreting and verifying what a $\partial \mathbb{B}$ net has learned is relatively less difficult. The option to bias towards learning boolean functions will reduce variance in some domains. And boolean-valued, 1-bit weights increase the memory-efficiency of trained models. -The main contributions of this work are (i) defining novel activation functions that harden to semantically equivalent boolean functions, (ii) defining novel network architectures that effectively learn boolean functions to solve multi-class classification problems, and (iii) experiments that demonstrate $\partial \mathbb{B}$ nets compete with existing approaches in terms of predictive performance yet yield considerably smaller models. +The main contributions of this work are (i) defining novel activation functions that `harden' to semantically equivalent boolean functions, (ii) defining novel network architectures to effectively learn boolean functions that solve multi-class classification problems, and (iii) experiments that demonstrate $\partial \mathbb{B}$ nets compete with existing approaches in terms of predictive performance yet yield considerably smaller models. This paper examines related work (section \ref{sec:related-work}), (ii) etc. @@ -216,6 +216,14 @@ \subsubsection{Implies} \subsubsection{Majority} +\begin{figure}[t] + \centering + \includegraphics[trim=0pt 0pt 0pt 0pt, clip, width=0.975\textwidth]{majority-gates.png} + \caption{{\em Majority.}} + \label{fig:majority-plot} +\end{figure} + + Define $\operatorname{majority-index}: \mathbb{Z}_{>0} \to \mathbb{Z}_{> 0}$ as $n \mapsto 1 + \lfloor \frac{n-1}{2}\rfloor$. %Define $\operatorname{select}: [0,1]^n \times {1, 2, \ldots, n} \to [0,1]$ as $({\bf x}, i) \mapsto x_{i}$. @@ -379,6 +387,7 @@ \subsection{MNIST} \section{Conclusion} +TODO: we can write boolean functions, soften them, and place them within differentiable neural networks. \subsubsection*{Acknowledgments} Use unnumbered third level headings for the acknowledgments. 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zIpUr^ryY=3*w{IoHs2Njti_*o-D3+bWqOhN!O-kT2%w>uNVs0*nAD@1CJ?)Au52?x*HfYV(jgkwN%1#?F<21ZZM rHMI1ya7UL#S{TQFslO5@oRaeAMw{0e+IGNC0pfwxb=AsMtit{Wi1!Io diff --git a/docs/db.tex b/docs/db.tex index d9c1d53..f74cdcf 100644 --- a/docs/db.tex +++ b/docs/db.tex @@ -11,7 +11,7 @@ \usepackage{graphicx} \usepackage{comment} -\title{$\partial\mathbb{B}$ nets learn boolean functions by\\backpropagation} +\title{$\partial\mathbb{B}$ nets: learning boolean functions by\\gradient descent} % Authors must not appear in the submitted version. They should be hidden % as long as the \iclrfinalcopy macro remains commented out below. @@ -43,17 +43,17 @@ \begin{abstract} $\partial\mathbb{B}$ nets are differentiable neural networks - that learn discrete boolean functions by backpropagation. - $\partial\mathbb{B}$ nets, once trained, `harden' to boolean functions with identical semantics. In consequence, the boolean functions have identical accuracy to the trained nets, unlike existing approaches to neural network binarization. Experiments demonstrate that $\partial\mathbb{B}$ nets achieve competitive performance on standard machine learning problems yet are significantly more compact (due to 1-bit weights) and interpretable (due to the logical nature of the learnt function). + that learn discrete boolean functions by gradient descent. + $\partial\mathbb{B}$ nets, once trained, `harden' to boolean functions with identical semantics. In consequence, the boolean functions have identical accuracy to the trained nets, unlike existing approaches to neural network binarization. Experiments demonstrate that $\partial\mathbb{B}$ nets achieve comparable performance on standard machine learning problems yet are significantly more compact (due to 1-bit weights) and interpretable (due to the logical nature of the learnt function). \end{abstract} \section{Introduction} -Neural networks are typically trained by gradient descent in weight-space, where the direction of descent minimises some loss function. The backpropagation algorithm \citep{rumelhart1986learning} is a highly efficient method to calculate the partial derivatives of the loss with respect to the network's weights. In consequence, typical neural networks are differentiable functions with weights represented by machine floats. This approach has led to tremendous advances in machine learning. +Typical neural networks are differentiable functions with weights represented by machine floats. Training consists of gradient descent in weight-space, where the direction of descent minimises loss and the gradient is efficiently calculated by the backpropagation algorithm \citep{rumelhart1986learning}. This approach has led to tremendous advances in machine learning. -However, there are drawbacks. First, we cannot easily learn discrete functions, such as logical predicates. In consequence, interpreting or verifying what a network has learned is difficult. Also, many learning problems naturally arise in boolean rather than real-valued domains. Second, representing weights as machine floats confers the benefit of time-efficient training but at the cost of memory-inefficient trained models. For example, network quantisation techniques (TODO:refs) demonstrate that full 64 of 32-bit precision weights are often unnecessary for a network's predictive performance, although there is a trade-off. +However, there are drawbacks. First, the differentiability requirement means we cannot easily learn discrete functions, such as logical predicates. In consequence, interpreting or verifying what a network has learned is difficult. Also, many learning problems naturally arise in discrete rather than continuous domains. Second, representing weights as machine floats enables time-efficient training but at the cost of memory-inefficient models. For example, network quantisation techniques (TODO:refs) demonstrate that full 64 of 32-bit precision weights are often unnecessary for final predictive performance, although there is a trade-off. -This paper proposes a new approach to mitigating these drawbacks. The main idea is to define a type of neural network, called a $\partial \mathbb{B}$ net, which has two aspects: a `soft' aspect, which is a differentiable real-valued function trainable by backpropagation, and a `hard' aspect, which is a non-differentiable boolean function. Both aspects are semantically equivalent. We train the `soft' net as normal, using backpropagation, then `harden' the network and its weights to yield a learned boolean function with equivalent predictive performance (see figure \ref{fig:main-idea}). In consequence, interpreting and verifying what a $\partial \mathbb{B}$ net has learned is relatively less difficult. The option to bias towards learning boolean functions will reduce variance in some domains. And boolean-valued, 1-bit weights increase the memory-efficiency of trained models. +This paper proposes a new approach to mitigating these drawbacks. The main idea is to define a type of neural network, called a $\partial \mathbb{B}$ net, which has two aspects: a soft net, which is a differentiable real-valued function, and a hard net, which is a non-differentiable, discrete function. Both aspects are semantically equivalent. We train the soft net as normal, using backpropagation, then `harden' the learned weights to boolean values and bind them with the hard net to yield a discrete function with identical predictive performance (see figure \ref{fig:main-idea}). In consequence, interpreting and verifying a $\partial \mathbb{B}$ net is relatively less difficult. The bias towards learning discrete functions reduces variance in some domains. And boolean-valued, 1-bit weights increase the memory-efficiency of trained models. The main contributions of this work are (i) defining novel activation functions that `harden' to semantically equivalent boolean functions, (ii) defining novel network architectures to effectively learn boolean functions that solve multi-class classification problems, and (iii) experiments that demonstrate $\partial \mathbb{B}$ nets compete with existing approaches in terms of predictive performance yet yield considerably smaller models. @@ -62,7 +62,7 @@ \section{Introduction} \begin{figure}[h] \centering \includegraphics[width=0.9\textwidth]{db-net.png} - \caption{{\em Learning discrete boolean-valued functions with a $\partial\mathbb{B}$ net.} A $\partial \mathbb{B}$ net specifies (i) a differentiable neural network that is hard-equivalent to (ii) a non-differentiable discrete function. The neural network is trained as normal with backpropagation to yield a set of real weights. The real weights are hardened to boolean values and then bound with the discrete function. The result is a learned discrete function that performs identically to the trained network.} + \caption{{\em Learning discrete functions with a $\partial\mathbb{B}$ net.} A $\partial \mathbb{B}$ net specifies (i) a differentiable neural network that is hard-equivalent to (ii) a non-differentiable discrete function. The neural network is trained as normal with backpropagation to yield a set of real weights. The real weights are hardened to boolean values and then bound with the discrete function. The result is a learned discrete function that performs identically to the trained network.} \label{fig:main-idea} \end{figure} @@ -71,10 +71,14 @@ \section{Related work}\label{sec:related-work} \section{$\partial\mathbb{B}$ nets} +A $\partial \mathbb{B}$ net has two aspects, a soft net and a hard net. Both nets use bits to represent transitory values and learnable weights. However, a soft net uses soft-bits and a hard net uses hard-bits. + \begin{definition}[Soft-bits and hard-bits] A {\em soft-bit} is a real value in the range $[0,1]$ and a {\em hard-bit} is a boolean value from the set $\{0,1\}$. A soft-bit, $x$, is {\em high} if $x>1/2$, otherwise it is {\em low}. \end{definition} +A hardening function converts soft-bits to hard-bits. + \begin{definition}[Hardening] The {\em hardening} function, $\operatorname{harden}(x_{1}, \dots, x_{n}) = [f(x_{1}), \dots, f(x_{n})]$, converts soft-bits to hard-bits, where \begin{equation*} @@ -86,21 +90,53 @@ \section{$\partial\mathbb{B}$ nets} \end{equation*} \end{definition} +The soft-bit value $1/2$ is therefore a threshold. Above this threshold the soft-bit represents $\text{True}$, otherwise it represents $\text{False}$. + +A soft net is any differentiable function, $f$, that `hardens' to a semantically equivalent discrete function, $g$. For example, if $f(x) = 1 - x$, where $x \in [0,1]$, and $g(y) = \neg y$, where $y \in \{0,1\}$ then: if $x$ is high (resp. low) then both $f(x)$ and $g(\operatorname{harden}(x))$ are low (resp. high). In other words, $f$ is hard-equivalent to boolean negation. More generally: + \begin{definition}[Hard-equivalence] - A function, $f: [0,1]^n \rightarrow [0,1]^m$, is {\em hard-equivalent} to a boolean function, $g: \{1,0\}^n \rightarrow \{1,0\}^m$, if + A function, $f: [0,1]^n \rightarrow [0,1]^m$, is {\em hard-equivalent} to a discrete function, $g: \{1,0\}^n \rightarrow \{1,0\}^m$, if \begin{equation*} \operatorname{harden}(f(\operatorname{harden}({\bf x}))) = g(\operatorname{harden}({\bf x})) \end{equation*} for all ${\bf x} \in [0,1]^{n}$. \end{definition} -\subsection{Differentiable boolean functions} +$\partial \mathbb{B}$ nets are arbitrary compositions of differentiable functions that are hard-equivalent to boolean functions (and natural generalisations). -Weights are trainable soft-bits. A high weight implies the corresponding operation is masked out and therefore inactive. +Neural networks are typically composed of nonlinear activation functions (for representational generality) that are strictly monotonic (so gradients always exist that link changes in inputs to outputs) and differentiable (so gradients reliably represent the local loss surface). Activation functions that are only monotonic (so some gradients are zero) and differentiable almost everywhere (so some gradients are undefined) can also work, e.g. RELU \citep{10.5555/3104322.3104425}. $\partial \mathbb{B}$ nets are also composed of `activation' functions that satisfy these properties. But $\partial \mathbb{B}$ net activation functions must also satisfy the additional requirement of hard-equivalence. -TODO: requirements are smooth almost everywhere, strictly increasing or decreasing (i.e. no zero gradient). We also want to maximise fan-in information where necessary (for backprop error signal) but also minimize it where necessary (e.g. picking a representative bit we want to optimize). Representative bit when we want a vector-wise update; otherwise, min/max. +\begin{figure}[t] + \centering + \includegraphics[trim=0pt 0pt 0pt 0pt, clip, width=0.975\textwidth]{logic-gates.png} + \caption{{\em Gradient-rich versus gradient-sparse differentiable boolean functions.} Each column contains contour plots of functions $f(x,y)$ that are hard-equivalent to a boolean function (one of $\neg(x \oplus y)$, $x \wedge y$, $x \vee y$, or $x \Rightarrow y$). Every function is continuous and differentiable almost everywhere (white lines indicate non-continuous derivatives). The upper plots are gradient-sparse, where vertical and horizontal contours indicate the function is constant with respect to one of its inputs, i.e. $\partial f/\partial y = 0$ or $\partial f/\partial x = 0$. The lower plots are gradient-rich, where the curved contours indicate the function always varies with respect to any of its inputs, i.e. $\partial f/\partial y \neq 0$ and $\partial f/\partial x \neq 0$. $\partial \mathbb{B}$ nets use gradient-rich functions to ensure that error is always backpropagated to all inputs.} + \label{fig:gradient-rich} +\end{figure} + +\subsection{Learning to negate} + +We want to learn whether to negate a boolean value, $x$, or simply leave it unaltered. We represent this decision by a boolean weight, $w$, where low $w$ means negate and high $w$ means do nothing. The boolean function that meets this requirement is $\neg(x \oplus w)$. However, this function is not differentiable. We therefore define the differentiable function, + \begin{equation*} + \begin{aligned} + \partial\text{NOT}: [0, 1]^{2} &\to [0,1], \\ + (w, x) &\mapsto 1 - w + x (2w - 1)\text{,} + \end{aligned} + \end{equation*} +which is hard-equivalent to the boolean function $\neg(x \oplus w)$ (see proposition \ref{prop:not}). + +Note that $\operatorname{min}$ is hard-equivalent to $\wedge$ and $\operatorname{max}$ is hard-equivalent to $\vee$. So the function $\operatorname{max}(\operatorname{min}(w, x), \operatorname{min}(1-w, 1-x))$ is also hard-equivalent to $\neg(x \oplus w)$. But $\partial\text{NOT}$, in contrast, is a gradient-rich function that always backpropagates error to all its inputs (see figure \ref{fig:gradient-rich}). + +\subsection{Differentiable $\wedge$} + +As stated $\operatorname{min}(x, y)$ is differentiable and hard-equivalent to $x \wedge y$. However, it is gradient-sparse, e.g. $\frac{\partial}{\partial x} \operatorname{min}(x,y) = 0$ when $(x,y)=(0.9,0.1)$. Other soft logic functions, such as $f(x,y) = x y$, are not hard-equivalent at intermediate soft-bit values, e.g. $f(0.6, 0.6) = 0.36$. To construct a gradient-rich, hard-equivalent function we first observe that $\operatorname{min}(x,y)$ essentially selects one of $x$ or $y$ as a `representative soft-bit' that is guaranteed to be hard-equivalent to $x \wedge y$. The representative bit, whether low or high, has a corresponding margin up to the threshold value $1/2$. + +\begin{figure}[t] + \centering + \includegraphics[width=1.0\textwidth]{margin-trick.png} + \caption{{\em todo}.} + \label{fig:margin-trick} +\end{figure} -TODO: min/max are a degenerate case of sorting/ordering Define \begin{equation*} @@ -122,29 +158,6 @@ \subsection{Differentiable boolean functions} \end{aligned} \end{equation*} -\begin{figure}[t] - \centering - \includegraphics[trim=0pt 0pt 0pt 0pt, clip, width=0.975\textwidth]{logic-gates.png} - \caption{{\em Gradient-rich versus gradient-sparse differentiable boolean functions.} Each column contains contour plots of functions $f(x,y)$ that are hard-equivalent to a boolean function (one of $\neg(x \oplus y)$, $x \wedge y$, $x \vee y$, or $x \Rightarrow y$). Every function is continuous and differentiable almost everywhere (white lines indicate non-continuous derivatives). The upper plots are gradient-sparse, where vertical and horizontal contours indicate the function is constant with respect to one of its inputs, i.e. $\partial f/\partial y = 0$ or $\partial f/\partial x = 0$. The lower plots are gradient-rich, where the curved contours indicate the function always varies with respect to any of its inputs, i.e. $\partial f/\partial y \neq 0$ and $\partial f/\partial x \neq 0$. $\partial \mathbb{B}$ nets use gradient-rich functions to ensure that error is always backpropagated to all inputs.} - \label{fig:and-plot} -\end{figure} - -\subsubsection{Not} - -TODO: this is not the not operator, but `learn not'. - -Define - \begin{equation*} - \begin{aligned} - \partial\text{NOT}: [0, 1]^{2} &\to [0,1], \\ - (w, x) &\mapsto 1 - w + x (2w - 1)\text{,} - \end{aligned} - \end{equation*} -where $w$ is a weight and $x$ is a soft-bit value. -$\partial${NOT} is hard-equivalent to the boolean function $\neg(x \oplus w)$ (proposition \ref{prop:not}). If the weight is high then $\partial${NOT} is hard-equivalent to the boolean identity function; otherwise it is hard-equivalent to $\neg$. In consequence, we can learn to logically not, or simply pass through, the input value $x$. - - -\subsubsection{And} Define \begin{equation*} @@ -166,7 +179,11 @@ \subsubsection{And} $\partial${AND} is hard-equivalent to the boolean function $x \wedge y$ (proposition \ref{prop:and}). -\subsubsection{Or} +We also want to maximise fan-in information where necessary (for backprop error signal) but also minimize it where necessary (e.g. picking a representative bit we want to optimize). Representative bit when we want a vector-wise update; otherwise, min/max. + +TODO: min/max are a degenerate case of sorting/ordering + +\subsection{Differentiable $\vee$} Define \begin{equation*} @@ -202,7 +219,7 @@ \subsubsection{Or} $\partial${OR} is hard-equivalent to the boolean function $x \vee y$ (proposition \ref{prop:or}). -\subsubsection{Implies} +\subsection{Differentiable $\Rightarrow$} Define \begin{equation*} @@ -214,7 +231,7 @@ \subsubsection{Implies} where $w$ is a weight and $x$ is a soft-bit value. $\partial${IMPLIES} is hard-equivalent to the boolean function $w \Rightarrow x$ (proposition \ref{prop:implies}). -\subsubsection{Majority} +\subsection{Differentiable majority} \begin{figure}[t] \centering @@ -257,7 +274,7 @@ \subsubsection{Majority} TODO: explain how we use sort to avoid space blow-up. TODO: discuss time-complexity of sort. -\subsubsection{Count} +\subsection{Differentiable integer count} Define \begin{equation*} @@ -407,20 +424,19 @@ \section{Proofs} $\partial${NOT} is hard-equivalent to the boolean function $\neg (x \oplus w)$. \begin{proof} - Table \ref{not-table} is the truth table of the boolean function $\neg (x \oplus w)$. + Table \ref{not-table} is the truth table of the boolean function $\neg (x \oplus w)$, where $h(x) = \operatorname{harden}(x)$. \begin{table} \begin{center} - \begin{tabular}{lll} - \multicolumn{1}{c}{$h(w)$} &\multicolumn{1}{c}{$h(x)$} &\multicolumn{1}{c}{$\partial\text{NOT}(h(w), h(x))$} + \begin{tabular}{cccccc} + \multicolumn{1}{c}{$w$} &\multicolumn{1}{c}{$x$} &\multicolumn{1}{c}{$h(w)$} &\multicolumn{1}{c}{$h(x)$} &\multicolumn{1}{c}{$\partial\text{NOT}(h(w), h(x))$} &\multicolumn{1}{c}{$h(\partial\text{NOT}(h(w), h(x)))$} \\ \hline \\ - 0 & 0 & 1\\ - 1 & 0 & 0\\ - 0 & 1 & 0\\ - 1 & 1 & 1\\ + $\left[0, \frac{1}{2}\right]$ & $\left[0, \frac{1}{2}\right]$ & 0 & 0 & 1 & 1\\[0.1cm] + $\left(\frac{1}{2}, 1\right]$ & $\left[0, \frac{1}{2}\right]$ &1 & 0 & 0 & 0\\[0.1cm] + $\left[0, \frac{1}{2}\right]$ & $\left(\frac{1}{2}, 1\right]$ &0 & 1 & 0 & 0\\[0.1cm] + $\left(\frac{1}{2}, 1\right]$ & $\left(\frac{1}{2}, 1\right]$ &1 & 1 & 1 & 1\\[0.1cm] \end{tabular} \end{center} \caption{$\partial${NOT} is hard-equivalent to $\neg (x \oplus w)$.}\label{not-table} - \end{table} \end{proof} \end{prop} @@ -429,63 +445,60 @@ \section{Proofs} \begin{prop}\label{prop:and} $\partial${AND} is hard-equivalent to the boolean function $x \wedge y$. \begin{proof} - Table \ref{and-table} is the truth table of the boolean function $x \wedge y$. + Table \ref{and-table} is the truth table of the boolean function $x \wedge y$, where $h(x) = \operatorname{harden}(x)$.. \begin{table} \begin{center} - \begin{tabular}{llll} - \multicolumn{1}{c}{$h(x)$} &\multicolumn{1}{c}{$h(y)$} &\multicolumn{1}{c}{$\partial\text{AND}(h(x), h(y))$} &\multicolumn{1}{c}{$h(\partial\text{AND}(h(x), h(y)))$} + \begin{tabular}{cccccc} + \multicolumn{1}{c}{$w$} &\multicolumn{1}{c}{$x$} &\multicolumn{1}{c}{$h(w)$} &\multicolumn{1}{c}{$h(x)$} &\multicolumn{1}{c}{$\partial\text{AND}(h(w), h(x))$} &\multicolumn{1}{c}{$h(\partial\text{AND}(h(w), h(x)))$} \\ \hline \\ - 1 & 1 & 1 & 1\\ - 1 & 0 & 1/4 & 0\\ - 0 & 1 & 1/4 & 0\\ - 0 & 0 & 0 & 0\\ + $\left[0, \frac{1}{2}\right]$ & $\left[0, \frac{1}{2}\right]$ & 0 & 0 & 0 & 0\\[0.1cm] + $\left(\frac{1}{2}, 1\right]$ & $\left[0, \frac{1}{2}\right]$ &1 & 0 & $\frac{1}{4}$ & 0\\[0.1cm] + $\left[0, \frac{1}{2}\right]$ & $\left(\frac{1}{2}, 1\right]$ &0 & 1 & $\frac{1}{4}$ & 0\\[0.1cm] + $\left(\frac{1}{2}, 1\right]$ & $\left(\frac{1}{2}, 1\right]$ &1 & 1 & 1 & 1\\[0.1cm] \end{tabular} \end{center} \caption{$\partial${AND} is hard-equivalent to $x \wedge y$.}\label{and-table} - - \end{table} + \end{table} \end{proof} \end{prop} \begin{prop}\label{prop:or} $\partial${OR} is hard-equivalent to the boolean function $x \vee y$. \begin{proof} - Table \ref{or-table} is the truth table of the boolean function $x \vee y$. + Table \ref{or-table} is the truth table of the boolean function $x \vee y$, where $h(x) = \operatorname{harden}(x)$.. \begin{table} - \begin{center} - \begin{tabular}{llll} - \multicolumn{1}{c}{$h(x)$} &\multicolumn{1}{c}{$h(y)$} &\multicolumn{1}{c}{$\partial\text{OR}(h(x), h(y))$} &\multicolumn{1}{c}{$h(\partial\text{OR}(h(x), h(y)))$} - \\ \hline \\ - 1 & 1 & 1 & 1\\ - 1 & 0 & 3/4 & 1\\ - 0 & 1 & 3/4 & 1\\ - 0 & 0 & 0 & 0\\ - \end{tabular} - \end{center} - \caption{$\partial${OR} is hard-equivalent to $x \vee y$.}\label{or-table} - - \end{table} + \begin{center} + \begin{tabular}{cccccc} + \multicolumn{1}{c}{$w$} &\multicolumn{1}{c}{$x$} &\multicolumn{1}{c}{$h(w)$} &\multicolumn{1}{c}{$h(x)$} &\multicolumn{1}{c}{$\partial\text{OR}(h(w), h(x))$} &\multicolumn{1}{c}{$h(\partial\text{OR}(h(w), h(x)))$} + \\ \hline \\ + $\left[0, \frac{1}{2}\right]$ & $\left[0, \frac{1}{2}\right]$ & 0 & 0 & 0 & 0\\[0.1cm] + $\left(\frac{1}{2}, 1\right]$ & $\left[0, \frac{1}{2}\right]$ &1 & 0 & $\frac{3}{4}$ & 1\\[0.1cm] + $\left[0, \frac{1}{2}\right]$ & $\left(\frac{1}{2}, 1\right]$ &0 & 1 & $\frac{3}{4}$ & 1\\[0.1cm] + $\left(\frac{1}{2}, 1\right]$ & $\left(\frac{1}{2}, 1\right]$ &1 & 1 & 1 & 1\\[0.1cm] + \end{tabular} + \end{center} + \caption{$\partial${OR} is hard-equivalent to $x \vee y$.}\label{or-table} + \end{table} \end{proof} \end{prop} \begin{prop}\label{prop:implies} $\partial${IMPLIES} is hard-equivalent to the boolean function $w \Rightarrow x$. \begin{proof} - Table \ref{implies-table} is the truth table of the boolean function $x \Rightarrow y$. + Table \ref{implies-table} is the truth table of the boolean function $x \Rightarrow y$, where $h(x) = \operatorname{harden}(x)$.. \begin{table} - \begin{center} - \begin{tabular}{llll} - \multicolumn{1}{c}{$h(w)$} &\multicolumn{1}{c}{$h(x)$} &\multicolumn{1}{c}{$\partial\text{IMPLIES}(h(w), h(x))$} &\multicolumn{1}{c}{$h(\partial\text{IMPLIES}(h(w), h(x)))$} - \\ \hline \\ - 1 & 1 & 3/4 & 1\\ - 1 & 0 & 0 & 0\\ - 0 & 1 & 1 & 1\\ - 0 & 0 & 3/4 & 1\\ - \end{tabular} - \end{center} - \caption{$\partial${IMPLIES} is hard-equivalent to $w \Rightarrow x$.}\label{implies-table} - - \end{table} + \begin{center} + \begin{tabular}{cccccc} + \multicolumn{1}{c}{$w$} &\multicolumn{1}{c}{$x$} &\multicolumn{1}{c}{$h(w)$} &\multicolumn{1}{c}{$h(x)$} &\multicolumn{1}{c}{$\partial\text{IMPLIES}(h(w), h(x))$} &\multicolumn{1}{c}{$h(\partial\text{IMPLIES}(h(w), h(x)))$} + \\ \hline \\ + $\left[0, \frac{1}{2}\right]$ & $\left[0, \frac{1}{2}\right]$ & 0 & 0 & $\frac{3}{4}$ & 1\\[0.1cm] + $\left(\frac{1}{2}, 1\right]$ & $\left[0, \frac{1}{2}\right]$ &1 & 0 & 0 & 0\\[0.1cm] + $\left[0, \frac{1}{2}\right]$ & $\left(\frac{1}{2}, 1\right]$ &0 & 1 & 1 & 1\\[0.1cm] + $\left(\frac{1}{2}, 1\right]$ & $\left(\frac{1}{2}, 1\right]$ &1 & 1 & $\frac{3}{4}$ & 1\\[0.1cm] + \end{tabular} + \end{center} + \caption{$\partial${IMPLIES} is hard-equivalent to $x \Rightarrow y$.}\label{implies-table} + \end{table} \end{proof} \end{prop} diff --git a/docs/majority-gates.png b/docs/majority-gates.png new file mode 100644 index 0000000000000000000000000000000000000000..8e72807d0c1996034a1c0640250536f1fafaa6f6 GIT binary patch literal 192547 zcmd43cQ}@RA3lE3unMIpnUz^aQe>p0NJ6qgC{ZMPXBDMkq>Mz!h{&kO%4ne!*~uz< z%ijFXt7m?Dj_>jN=l93&K91+{ad*3}_w|0i#`!v5=j*<$uBy16mX($ui1o*n6f_8e z+KeDp2U1hw9ZFsifBdL+)6{j)IOoh|V`ptRZB?>uCSxpDM@i>RdKkeW|vTJ+9_Mm&$N5 z)x9?@kud{YvWxVoi{k#DT=N#J9^@TdyFrb6j54dQ$%cMowCBjgn<&?j>^3UdOvCLf za;cK~FNW@8$cURPD}9+`%;HPLYK-%2n=%eZ zbsbJyn>je2voj@**qNSlFulO#Z0>N0%i)qKmyWfinccZ7Tq@_RE;^pOXv!tU zEJ1J)#}y81xYv&$6?CiPmbc|2U#)N4)Gg)bzppySz{tf|er(!dW+E=iZN9Q5LHcp}$Uytd zUXk?lqLhIPb*Gc&>*DroSx27npI1p<9zXlPej&WaTSDlGeexNNz1`h3+%2uGvNAFu z^i+g#UR+#UN5^??-;9imWAUe?*a+_%>(;)1@5iu)M~aX7cbIrKiYTsSjB3BE&Q$VJ>^Ng zY>OK=Zwj*06D4-I0t1H(qr8HG!Z$Gr!jCt!uc)YKYM}l>*u4gdf35dY1YzSoGupng 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Y7=NIjc60SxJ8$D>>!`tGcqig%F>nx6nk1fwr zRG?e?z2A4-1)UGGu_a(*v`J)NpijZ8l6d>}{cb0T7&ik7r60)z;^Joktf8ySOi!PX zj>dUGY5`!;d5}P8Yw61TDvU^|W&NOrquV0$i2~HVJx4xUo0;)%*^<>efoD3u@FgWt zq|JFX+K>STdux=dFR!dz(LE#^H8a4Ho}OO*LGNJzv##dnD(>W*rF@r|-y-BP7n$Hu z23_6UydxEKhwVwR1PxDVA{tI9%iQ944jLPI5v0D8lM^2+tPDbZ#|?ykCDNpjErXQCPobM?T@gW})D5OEl=>uI+r<=X~J z1xm?-UQchjF$HNZAlgq;C=?`vgqw46NaIaPikyl-ay#u;BUr z%>;P%Rdlz1Z`?714=&G6ow|T)L@5Ms(LcK8O9*#`_~<9{RaaNV%xGfR#fFr`ulFmG zMAir$#f6TOca_n~VI{p`2FB&vH&E)H*%CcEvxw5Y3OE1I?@K+8a&rD*&O*4r1qCT- z1S8erWyS@bw=an)g5+5Yxu6pA8Z0DQFU%#A&b_^1>A% zkDnt%3z~IkXz2TXEhD&Y354N99PC_XXCaX|d&S5aeO(j9!Q9&F>ZN66SD}u=VUo2~ zC+Y5UWau>=?qnClrU4%D2;n>VS(#>I{$12|^_ z{QNM8Gf?ZyPM+gP)2#VHfHVC7V>ddE)pFKgQc@C5-0WEA`ng_Gl{$irF{JSvM&i1`7?@TQ%7k|`m=i0g_r*?O!WVNki>qeh= z(%WvpO_J0&ZXuzoKnAuNV<;HwFD-~(|M8jYE02dW_)AwIwvja}onhV&aphZRhT#SH z-eYyC$a_U}4dXFxwbQmKo~a3D+l=M(tkw8E$(b_9pg|}v7f>aMriO+sQ1+z@pvK_AlsKijHB%4K0Zh?%nfFZ`U#4y=Lb`HWQve<7&V0)rL1jk8K?o}@%> zNk!Ha^SX6e-A=hDCk6-Ghh~DVFZ#rPYGoto-mRD$@WG+f0fz{214rh2En6f3X%k3u zNS2PBg~AcKX-0lWoctQnd9k|;GrXyEH<&3GpWCmCZA-woU=7<8q$q~6IxBc{Y-SOj zvd7)ykS`W`f5!dN)y(~2#I`qpyaoxk=)9{TXwY!{xsl3O*Bbu3YL)@fzx{~;Q&3=F2&b@9Z||LB&@tQC0M^buwrg8= z2q#-Ml{EPk6B7{RE}F{hZ~2p!mQa#s_f7yM>vAfxIo-4=sQP~A7!bBeY)Q9SJkFhR zSZjAlSeBBXmnXM-w>#waS*DRr`k}#unIvhp=K;2rjxmowQJRX7f$Dwn?8-)F6WjwP zk|dHb6XSjBn*V)l%oJ8i`p=utH7VZzd`hCm|MxRHdEWC98v@^r)J$%5JlBZ`E_ Date: Wed, 22 Mar 2023 15:25:19 +0000 Subject: [PATCH 050/113] more --- docs/db.tex | 45 +++++++++++++++++++++++---------------------- 1 file changed, 23 insertions(+), 22 deletions(-) diff --git a/docs/db.tex b/docs/db.tex index 4085894..7d39dd6 100644 --- a/docs/db.tex +++ b/docs/db.tex @@ -10,6 +10,8 @@ \usepackage{amsthm} \usepackage{graphicx} \usepackage{comment} +\usepackage{algorithm} +\usepackage{algpseudocode} \title{$\partial\mathbb{B}$ nets: learning boolean functions by\\gradient descent} @@ -126,7 +128,7 @@ \subsection{Learning to negate} \subsection{Margin packing} -Say we aim to construct a differentiable analogue of $x \wedge y$. Note that $\operatorname{min}(x,y)$ essentially selects one of $x$ or $y$ as a representative soft-bit that is guaranteed hard-equivalent to $x \wedge y$. However, by selecting only one of $x$ or $y$ then $\operatorname{min}$ is also guaranteed to be gradient-sparse. We define a `margin packing' method to avoid this dilemma. +Say we aim to construct a differentiable analogue of $x \wedge y$. Note that $\operatorname{min}(x,y)$ essentially selects one of $x$ or $y$ as a representative soft-bit that is guaranteed hard-equivalent to $x \wedge y$. However, by selecting only one of $x$ or $y$ then $\operatorname{min}$ is also guaranteed to be gradient-sparse. We define a `margin packing' method to solve this dilemma. The main idea of margin packing is (i) select a representative bit that is hard-equivalent to the target discrete function, and then (ii) pack a fraction of the margin between the representative bit and the hard threshold $1/2$ with gradient-rich information. The result is an augmented bit that is a function of all inputs yet hard-equivalent to the target function. @@ -159,7 +161,7 @@ \subsection{Margin packing} \end{aligned} \end{equation*} If the representative bit is high (resp. low) then the augmented bit is also high (resp. low). -But the augmented bit has a higher (resp. lower) value than the representative bit when below (resp. above) the $1/2$ threshold. The difference depends on the size of the available margin and the mean soft-bit value. Almost everywhere, an increase (resp. decrease) of the mean soft-bit increases (resp. decreases) the value of the augmented bit (see figure \ref{fig:margin-trick}). Note that if the $i$th bit is representative (i.e. hard-equivalent to the target function) then so is the augmented bit (see proposition \ref{prop:augmented}). We now use margin packing to define gradient-rich, hard-equivalents of boolean functions. +But the augmented bit has a higher (resp. lower) value than the representative bit when below (resp. above) the $1/2$ threshold. The difference depends on the size of the available margin and the mean soft-bit value. Almost everywhere, an increase (resp. decrease) of the mean soft-bit increases (resp. decreases) the value of the augmented bit (see figure \ref{fig:margin-trick}). Note that if the $i$th bit is representative (i.e. hard-equivalent to the target function) then so is the augmented bit (see proposition \ref{prop:augmented}). We use margin packing, where appropriate, to define gradient-rich, hard-equivalents of boolean functions. \subsection{Differentiable $\wedge$} @@ -228,7 +230,7 @@ \subsection{Differentiable $\Rightarrow$} \subsection{Differentiable majority} -\begin{figure}[t] +\begin{figure}[h] \centering \includegraphics[trim=0pt 0pt 0pt 0pt, clip, width=1.0\textwidth]{majority-gates.png} \caption{{\em Differentiable boolean majority.} The boolean majority function for three variables in DNF form is $\operatorname{Maj}(x,y,z) = (x \wedge y) \vee (x \wedge y) \vee (y \wedge z)$. The upper row contains contour plots of $f(x,y,z) = \operatorname{min}(\operatorname{max}(x,y), \operatorname{max}(x,z), \operatorname{max}(y,z))$ for values of $z \in \{0.2, 0.4, 0.6, 0.8\}$. $f$ is differentiable and $\equiv \operatorname{Maj}$ but gradient-sparse (vertical and horizontal contours indicate constancy with respect to an input). Also, the number of terms in $f$ grows exponentially with the number of variables. The lower row contains contour plots of $\partial\!\operatorname{Maj}(x,y,z)$ for the same values of $z$. $\partial\!\operatorname{Maj}$ is differentiable and $\equiv \operatorname{Maj}$ yet gradient-rich (curved contours indicate variability with respect to any inputs). In addition, the number of terms in $\partial\!\operatorname{Maj}$ is constant with respect to the number of variables.} @@ -247,9 +249,9 @@ \subsection{Differentiable majority} \text{,} \end{aligned} \end{equation*} -Interpret each input bit $x_{i}$ as a vote, yes or no, for a binary decision. If the majority of voters are in favour then $\operatorname{Maj}$ outputs 1. The majority function, in the context of predictive model, therefore aggregates multiple bits of weak evidence into a hard decision. Neural network binarization transforms real-valued activation functions into boolean threshold functions \citep{10.5555/3157382.3157557}, which indicates their close association. We aim to construct a differentiable analogue of $\operatorname{Maj}$. +Interpret each input bit $x_{i}$ as a vote, yes or no, for a binary decision. If the majority of voters are in favour then $\operatorname{Maj}$ outputs 1. The majority function, in the context of a predictive model, aggregates multiple bits of weak evidence into a hard decision. Neural network binarization transforms real-valued activation functions into boolean threshold functions \citep{10.5555/3157382.3157557}, which indicates their close association. We aim to construct a differentiable analogue of $\operatorname{Maj}$. -$\operatorname{Maj}$ for $n$ bits in DNF form is a disjunction of $\binom{n}{k}$ conjunctive clauses of size $k$, where $k=\lceil n/2 \rceil$ and each clause checks if a unique combination of a majority of the $n$ bits are high; e.g. $\operatorname{Maj}(x, y, z) = (x \wedge y) \vee (x \wedge y) \vee (y \wedge z)$. Therefore, we could in principle implement a differentiable analogue of $\operatorname{Maj}$ in terms of $\partial\wedge$ and $\partial\vee$. However, the number of terms grows exponentially (e.g. $n=50$ generates over 100 trillion conjunctive clauses, which is infeasible) and there is no known general algorithm for finding the minimal representation of $\operatorname{Maj}$ for arbitrary $n$. +$\operatorname{Maj}$ for $n$ bits in DNF form is a disjunction of $\binom{n}{k}$ conjunctive clauses of size $k$, where $k=\lceil n/2 \rceil$ and each clause checks whether a unique combination of a majority of the $n$ bits are all high; e.g. $\operatorname{Maj}(x, y, z) = (x \wedge y) \vee (x \wedge y) \vee (y \wedge z)$. Therefore, we could in principle implement a differentiable analogue of $\operatorname{Maj}$ in terms of $\partial\wedge$ and $\partial\vee$. However, the number of terms grows exponentially with the number of variables (e.g. $n=50$ generates over 100 trillion clauses, which is infeasible). And there is no known general algorithm for finding the minimal representation of $\operatorname{Maj}$ for arbitrary $n$. Instead, we trade-off time for memory costs. Observe that if the function $\operatorname{sort}({\bf x})$ sorts the elements of ${\bf x}$ in ascending order then the `middle' soft-bit is representative. For example, if ${\bf x} = [0.4, 0.9, 0.2]$ then $\operatorname{sort}({\bf x}) = [0.2, 0.4, 0.9]$ and the `middle' bit $x_{2}=0.4$ is low, which is hard-equivalent to $\operatorname{Maj}(0, 1, 0) = 0$. Define the index of the `middle' bit by \begin{equation*} @@ -266,29 +268,32 @@ \subsection{Differentiable majority} {\bf x} &\mapsto \operatorname{augmented-bit}(\operatorname{sort}({\bf x}), \operatorname{majority-index}({\bf x}))\text{,} \end{aligned} \end{equation*} -which is hard-equivalent to $\operatorname{Maj}({\bf x})$ (see proposition \ref{prop:majority}). Note that $\partial\!\operatorname{Maj}$ is differentiable almost everywhere and gradient-rich (see figure \ref{fig:majority-plot}). If $\operatorname{sort}$ is quicksort then the the average time-complexity of $\partial\!\operatorname{Maj}$ is $\mathcal{O}(n\log{}n)$, which is more expensive than $\partial\neg$, $\partial\wedge$, $\partial\vee$ and $\partial\!\Rightarrow$. +which is hard-equivalent to $\operatorname{Maj}({\bf x})$ (see proposition \ref{prop:majority}). Note that $\partial\!\operatorname{Maj}$ is differentiable almost everywhere and gradient-rich (see figure \ref{fig:majority-plot}). If $\operatorname{sort}$ is quicksort then the the average time-complexity of $\partial\!\operatorname{Maj}$ is $\mathcal{O}(n\log{}n)$, which makes $\partial\!\operatorname{Maj}$ more expensive than $\partial\neg$, $\partial\wedge$, $\partial\vee$ and $\partial\!\Rightarrow$. -\subsection{Differentiable integer count} +\subsection{Differentiable counting} -Define +A boolean counting function $f({\bf x})$ has the value $1$ if a counting predicate, $c({\bf x})$, holds for its $n$ inputs. For example, $\partial\!\operatorname{Maj}({\bf x})$ is a boolean counting function where $c({\bf x}) := |\{x_{i} : x_{i} = 1 \}| \geq \lceil \frac{n}{2} \rceil$. We aim to construct a differentiable analogue of $\operatorname{count}({\bf x}, k)$ where $c({\bf x}) := |\{x_{i} : x_{i} = 1 \}| = k$ (i.e. `exactly $k$ high'). This will be useful for multiclass classification problems where we interpret $k$ as a class prediction. + +As before, we use $\operatorname{sort}$ to trade-off time for memory costs. Observe that if the elements of ${\bf x}$ are in ascending order then, if any soft-bits are high, there exists a unique contiguous pair of indices $(i,i+1)$ where $x_{i}$ is low and $x_{i+1}$ is high, where index $i$ is a direct count of the number of soft-bits that are low in ${\bf x}$. In consequence, define \begin{equation*} \begin{aligned} -\operatorname{low-high}: [0,1]^{n} &\to [0,1]^{n+1},\\ -{\bf x} &\mapsto \left[ \operatorname{min}(1, x_{1}), \operatorname{min}(1 - x_{1}, x_{2}), \dots, \operatorname{min}(1 - x_{n-1}, x_{n}), \operatorname{min}(x_{n}, 1) \right]\text{.} +\partial\!\operatorname{count-hot}: [0,1]^{n} &\to [0,1]^{n+1}, \\ +{\bf x} &\mapsto \operatorname{low-high}(\operatorname{sort}({\bf x}))\text{,} \end{aligned} \end{equation*} - -Define +where \begin{equation*} \begin{aligned} -\partial\text{COUNT}: [0,1]^{n} &\to [0,1]^{n+1}, \\ -{\bf x} &\mapsto \operatorname{low-high}(\operatorname{sort}({\bf x}))\text{,} +\operatorname{low-high}: [0,1]^{n} &\to [0,1]^{n+1},\\ +{\bf x} &\mapsto \left[ \partial\!\wedge\!(1, x_{1}), \partial\!\wedge\!(1 - x_{1}, x_{2}), \dots, \partial\!\wedge\!(1 - x_{n-1}, x_{n}), \partial\!\wedge\!(1-x_{n}, 1) \right]\text{.} \end{aligned} \end{equation*} -where ${\bf x}$ is a vector of soft-bits. $\partial${COUNT} is hard-equivalent to +$\partial\!\operatorname{count-hot}({\bf x})$ outputs a 1-hot vector where the index of high bit is the number of low bits in ${\bf x}$. For example, $\partial\!\operatorname{count-hot}([0.1, 0.9, 0.2]) = [0.1, 0.2, \bold{0.8}, 0.1]\text{,}$ indicating that 2 bits are low, and $\partial\!\operatorname{count-hot}([0.6, 0.9, 0.7]) = [\bold{0.6}, 0.4, 0.3, 0.1]\text{,}$ indicating that 0 bits are low. + +Note that $\partial\!\operatorname{count-hot}$ is hard-equivalent to the boolean function \begin{equation*} \begin{aligned} -\text{COUNT}: \{0,1\}^{n} &\to \{0,1\}^{n+1}, \\ +\operatorname{count-hot}: \{0,1\}^{n} &\to \{0,1\}^{n+1}, \\ {\bf x} &\mapsto \left[\operatorname{k-of-n}({\bf x}, 0), \operatorname{k-of-n}({\bf x}, 1), \dots, \operatorname{k-of-n}({\bf x}, n)\right]\text{,} \end{aligned} \end{equation*} @@ -296,13 +301,9 @@ \subsection{Differentiable integer count} \begin{equation*} \operatorname{k-of-n}({\bf x}, k) = \bigvee_{|S|=k} \bigwedge_{i\in S} x_i \bigwedge_{j\notin S} \neg x_j \end{equation*} -(see proposition \ref{prop:count}). - -%$f(x_1, x_2, \ldots, x_n) = \bigvee_{|S|=k} \left( \bigwedge_{i\in S} x_i \bigwedge_{j\notin S} \neg x_j \right)$ -%is a logical OR that is taken over all possible subsets $S$ of size $k$ of the set ${1,2,\ldots,n}$. +(see proposition \ref{prop:count}). However, in the hardened aspect of the $\partial\mathbb{B}$ net we efficiently implement $\operatorname{count-hot}$ as a discrete program that simply counts the number of low bits. -TODO: count is 1-hot function -TODO: count allows us to implement boolean counting functions. +Note that $\partial\!\operatorname{count-hot}$ is differentiable and gradient-rich. We may construct various kinds of boolean counting functions from $\partial\!\operatorname{count-hot}$. For example, $\partial\!\operatorname{count}({\bf x}, k)$ is straightforwardly $\partial\!\operatorname{count-hot}({\bf x})[k]$. \subsection{Logical layers} From 47f177a31fa890388f4cb320e5055748c831e04a Mon Sep 17 00:00:00 2001 From: Ian Wright Date: Wed, 22 Mar 2023 21:49:54 +0000 Subject: [PATCH 051/113] more --- docs/db.tex | 111 +++++++++++++++++++++++++--------------------------- 1 file changed, 54 insertions(+), 57 deletions(-) diff --git a/docs/db.tex b/docs/db.tex index 7d39dd6..b2ae440 100644 --- a/docs/db.tex +++ b/docs/db.tex @@ -13,7 +13,7 @@ \usepackage{algorithm} \usepackage{algpseudocode} -\title{$\partial\mathbb{B}$ nets: learning boolean functions by\\gradient descent} +\title{$\partial\mathbb{B}$ nets: learning discrete functions by\\gradient descent} % Authors must not appear in the submitted version. They should be hidden % as long as the \iclrfinalcopy macro remains commented out below. @@ -57,7 +57,7 @@ \section{Introduction} This paper proposes a new approach to mitigating these drawbacks. The main idea is to define a type of neural network, called a $\partial \mathbb{B}$ net, which has two aspects: a soft net, which is a differentiable real-valued function, and a hard net, which is a non-differentiable, discrete function. Both aspects are semantically equivalent. We train the soft net as normal, using backpropagation, then `harden' the learned weights to boolean values and bind them with the hard net to yield a discrete function with identical predictive performance (see figure \ref{fig:main-idea}). In consequence, interpreting and verifying a $\partial \mathbb{B}$ net is relatively less difficult. The bias towards learning discrete functions reduces variance in some domains. And boolean-valued, 1-bit weights increase the memory-efficiency of trained models. -The main contributions of this work are (i) defining novel activation functions that `harden' to semantically equivalent boolean functions, (ii) defining novel network architectures to effectively learn boolean functions that solve multi-class classification problems, and (iii) experiments that demonstrate $\partial \mathbb{B}$ nets compete with existing approaches in terms of predictive performance yet yield considerably smaller models. +The main contributions of this work are (i) defining novel activation functions that `harden' to semantically equivalent discrete functions, (ii) defining novel network architectures to effectively learn discrete functions that solve multi-class classification problems, and (iii) experiments that demonstrate $\partial \mathbb{B}$ nets compete with existing approaches in terms of predictive performance yet yield considerably smaller models. This paper examines related work (section \ref{sec:related-work}), (ii) etc. @@ -118,13 +118,13 @@ \subsection{Learning to negate} We aim to learn to negate a boolean value, $x$, or simply leave it unaltered. Represent this decision by a boolean weight, $w$, where low $w$ means negate and high $w$ means do nothing. The boolean function that meets this requirement is $\neg(x \oplus w)$. However, this function is not differentiable. We therefore define the differentiable function, \begin{equation*} \begin{aligned} - \partial \neg: [0, 1]^{2} &\to [0,1], \\ + \partial_{\neg}: [0, 1]^{2} &\to [0,1], \\ (w, x) &\mapsto 1 - w + x (2w - 1)\text{,} \end{aligned} \end{equation*} -where $\partial \neg(w, x) \equiv \neg(x \oplus w)$ (see proposition \ref{prop:not}). +where $\partial_{\neg}(w, x) \equiv \neg(x \oplus w)$ (see proposition \ref{prop:not}). -Product logics, where for example $f(x,y) = x y$ is as a soft version of $x \wedge y$, although hard-equivalent at extreme values, e.g. $f(1,1)=1$ and $f(0,1)=0$, are not hard-equivalent at intermediate values, e.g. $f(0.6, 0.6) = 0.36$. G\"{o}del-style $\operatorname{min}$ and $\operatorname{max}$ functions, although hard-equivalent over the entire soft-bit range, i.e. $\operatorname{min}(x,y) \equiv x \wedge y$ and $\operatorname{min}(x,y) \equiv x \vee y$, are gradient-sparse in the sense that their outputs are not always a function of all their inputs, e.g. $\frac{\partial}{\partial x} \operatorname{max}(x,y) = 0$ when $(x,y)=(0.1, 0.9)$. So although the composite function $\operatorname{max}(\operatorname{min}(w, x), \operatorname{min}(1-w, 1-x))$ is differentiable and $\equiv \neg(x \oplus w)$ it does not always backpropagate error to its inputs. In contrast, $\partial \neg$ is a gradient-rich function that always backpropagates error to its inputs (see figure \ref{fig:gradient-rich}). +Product logics, where for example $f(x,y) = x y$ is as a soft version of $x \wedge y$, although hard-equivalent at extreme values, e.g. $f(1,1)=1$ and $f(0,1)=0$, are not hard-equivalent at intermediate values, e.g. $f(0.6, 0.6) = 0.36$. G\"{o}del-style $\operatorname{min}$ and $\operatorname{max}$ functions, although hard-equivalent over the entire soft-bit range, i.e. $\operatorname{min}(x,y) \equiv x \wedge y$ and $\operatorname{min}(x,y) \equiv x \vee y$, are gradient-sparse in the sense that their outputs are not always a function of all their inputs, e.g. $\frac{\partial}{\partial x} \operatorname{max}(x,y) = 0$ when $(x,y)=(0.1, 0.9)$. So although the composite function $\operatorname{max}(\operatorname{min}(w, x), \operatorname{min}(1-w, 1-x))$ is differentiable and $\equiv \neg(x \oplus w)$ it does not always backpropagate error to its inputs. In contrast, $\partial_{\neg}$ is a gradient-rich function that always backpropagates error to its inputs (see figure \ref{fig:gradient-rich}). \subsection{Margin packing} @@ -134,7 +134,7 @@ \subsection{Margin packing} \begin{figure}[t] \centering - \includegraphics[width=1.0\textwidth]{margin-trick.png} + \includegraphics[trim=30pt 5pt 30pt 10pt, clip, width=1.0\textwidth]{margin-trick.png} \caption{{\em Margin packing for constructing gradient-rich, hard-equivalent functions}. A representative bit, $z$, is hard-equivalent to a discrete target function but gradient-sparse (e.g. $z=\operatorname{min}(x,y) \equiv x \wedge y$). On the left $z$ is low, $z<1/2$; on the right $z$ is high, $z>1/2$. We can pack a fraction of the margin between $z$ and the hard threshold $1/2$ with additional gradient-rich information without affecting hard-equivalence. A natural choice is the mean soft-bit, $\bar{\bf x} \in [0,1]$. The grey shaded areas denote the packed margins and the final augmented bit. On the left $\approx 60\%$ of the margin is packed; on the right $\approx 90\%$.} \label{fig:margin-trick} \end{figure} @@ -168,36 +168,36 @@ \subsection{Differentiable $\wedge$} We aim to construct a differentiable analogue of the boolean function $\bigwedge_{i=1}^{n} x_i$. A representative bit is $\operatorname{min}(x_{1},\dots,x_{n})$. The function \begin{equation*} \begin{aligned} -\partial \wedge: [0,1]^{n} &\to [0,1], \\ +\partial_{\wedge}: [0,1]^{n} &\to [0,1], \\ {\bf x} &\mapsto \operatorname{augmented-bit}({\bf x}, \operatorname{argmin}\limits_{i} x[i]) \end{aligned} \end{equation*} is therefore hard-equivalent to the boolean function $\bigwedge_{i=1}^{n} x_i$ (see proposition \ref{prop:and}). In the special case $n=2$ we get the piecewise function, \begin{equation*} -\partial\!\wedge\!(x, y) = +\partial_{\wedge}\!(x, y) = \begin{cases} 1/2 + 1/2(x + y)(\operatorname{min}(x,y) - 1/2) & \text{if } \operatorname{min}(x,y) > 1/2 \\ \operatorname{min}(x,y) + 1/2(x + y)(1/2 - \operatorname{min}(x,y)) & \text{otherwise.} \end{cases} \end{equation*} -Note that $\partial \wedge$ is differentiable almost everywhere and gradient-rich (see figure \ref{fig:gradient-rich}). +Note that $\partial_{\wedge}$ is differentiable almost everywhere and gradient-rich (see figure \ref{fig:gradient-rich}). \subsection{Differentiable $\vee$} The differentiable analogue of $\vee$ is identical to $\wedge$, except the representative bit is selected by $\operatorname{max}$. The function \begin{equation*} \begin{aligned} -\partial\vee: [0,1]^{n} &\to [0,1], \\ +\partial_{\vee}: [0,1]^{n} &\to [0,1], \\ {\bf x} &\mapsto \operatorname{augmented-bit}({\bf x}, \operatorname{argmax}\limits_{i} x[i]) \end{aligned} \end{equation*} -is hard-equivalent to the boolean function $\bigvee_{i=1}^{n} x_i$ (see proposition \ref{prop:or}). Note that $\partial \vee$ is differentiable almost everywhere and gradient-rich (see figure \ref{fig:gradient-rich}). +is hard-equivalent to the boolean function $\bigvee_{i=1}^{n} x_i$ (see proposition \ref{prop:or}). Note that $\partial_{\vee}$ is differentiable almost everywhere and gradient-rich (see figure \ref{fig:gradient-rich}). \begin{comment} Define \begin{equation*} \begin{aligned} -\partial\!\vee\!(x, y) = +\partial_{\vee}\!(x, y) = \begin{cases} 1/2 + 1/2(x + y)(\operatorname{max}(x,y) - 1/2) & \text{if } \operatorname{max}(x,y) > 1/2 \\ \operatorname{max}(x,y) + 1/2(x + y)(1/2 - \operatorname{max}(x,y)) & \text{otherwise.} @@ -207,7 +207,7 @@ \subsection{Differentiable $\vee$} \begin{comment} \begin{equation*} \begin{aligned} -\partial \vee: [0,1]^{2} &\to [0,1], \\ +\partial_{\vee}: [0,1]^{2} &\to [0,1], \\ (x, y) &\mapsto \begin{cases} 1/2 + 1/2(x + y)(m - 1/2) & \text{if } 2m > 1 \\ @@ -219,11 +219,11 @@ \subsection{Differentiable $\vee$} \subsection{Differentiable $\Rightarrow$} -The differentiable analogue of $\Rightarrow$ is defined in terms of $\partial\vee$. The function +The differentiable analogue of $\Rightarrow$ is defined in terms of $\partial_{\vee}$. The function \begin{equation*} \begin{aligned} -\partial\!\Rightarrow: [0,1]^{2} &\to [0,1],\\ -(x, y) &\mapsto \partial\!\vee\!(y, 1-x)\text{,} +\partial_{\Rightarrow}: [0,1]^{2} &\to [0,1],\\ +(x, y) &\mapsto \partial_{\vee}\!(y, 1-x)\text{,} \end{aligned} \end{equation*} is hard-equivalent to $x \Rightarrow y$ (proposition \ref{prop:implies}). We can define analogues of all the basic boolean operators in a similar manner. @@ -251,7 +251,7 @@ \subsection{Differentiable majority} \end{equation*} Interpret each input bit $x_{i}$ as a vote, yes or no, for a binary decision. If the majority of voters are in favour then $\operatorname{Maj}$ outputs 1. The majority function, in the context of a predictive model, aggregates multiple bits of weak evidence into a hard decision. Neural network binarization transforms real-valued activation functions into boolean threshold functions \citep{10.5555/3157382.3157557}, which indicates their close association. We aim to construct a differentiable analogue of $\operatorname{Maj}$. -$\operatorname{Maj}$ for $n$ bits in DNF form is a disjunction of $\binom{n}{k}$ conjunctive clauses of size $k$, where $k=\lceil n/2 \rceil$ and each clause checks whether a unique combination of a majority of the $n$ bits are all high; e.g. $\operatorname{Maj}(x, y, z) = (x \wedge y) \vee (x \wedge y) \vee (y \wedge z)$. Therefore, we could in principle implement a differentiable analogue of $\operatorname{Maj}$ in terms of $\partial\wedge$ and $\partial\vee$. However, the number of terms grows exponentially with the number of variables (e.g. $n=50$ generates over 100 trillion clauses, which is infeasible). And there is no known general algorithm for finding the minimal representation of $\operatorname{Maj}$ for arbitrary $n$. +$\operatorname{Maj}$ for $n$ bits in DNF form is a disjunction of $\binom{n}{k}$ conjunctive clauses of size $k$, where $k=\lceil n/2 \rceil$ and each clause checks whether a unique combination of a majority of the $n$ bits are all high; e.g. $\operatorname{Maj}(x, y, z) = (x \wedge y) \vee (x \wedge y) \vee (y \wedge z)$. Therefore, we could in principle implement a differentiable analogue of $\operatorname{Maj}$ in terms of $\partial_{\wedge}$ and $\partial_{\vee}$. However, the number of terms grows exponentially with the number of variables (e.g. $n=50$ generates over 100 trillion clauses, which is infeasible). And there is no known general algorithm for finding the minimal representation of $\operatorname{Maj}$ for arbitrary $n$. Instead, we trade-off time for memory costs. Observe that if the function $\operatorname{sort}({\bf x})$ sorts the elements of ${\bf x}$ in ascending order then the `middle' soft-bit is representative. For example, if ${\bf x} = [0.4, 0.9, 0.2]$ then $\operatorname{sort}({\bf x}) = [0.2, 0.4, 0.9]$ and the `middle' bit $x_{2}=0.4$ is low, which is hard-equivalent to $\operatorname{Maj}(0, 1, 0) = 0$. Define the index of the `middle' bit by \begin{equation*} @@ -268,7 +268,7 @@ \subsection{Differentiable majority} {\bf x} &\mapsto \operatorname{augmented-bit}(\operatorname{sort}({\bf x}), \operatorname{majority-index}({\bf x}))\text{,} \end{aligned} \end{equation*} -which is hard-equivalent to $\operatorname{Maj}({\bf x})$ (see proposition \ref{prop:majority}). Note that $\partial\!\operatorname{Maj}$ is differentiable almost everywhere and gradient-rich (see figure \ref{fig:majority-plot}). If $\operatorname{sort}$ is quicksort then the the average time-complexity of $\partial\!\operatorname{Maj}$ is $\mathcal{O}(n\log{}n)$, which makes $\partial\!\operatorname{Maj}$ more expensive than $\partial\neg$, $\partial\wedge$, $\partial\vee$ and $\partial\!\Rightarrow$. +which is hard-equivalent to $\operatorname{Maj}({\bf x})$ (see proposition \ref{prop:majority}). Note that $\partial\!\operatorname{Maj}$ is differentiable almost everywhere and gradient-rich (see figure \ref{fig:majority-plot}). If $\operatorname{sort}$ is quicksort then the the average time-complexity of $\partial\!\operatorname{Maj}$ is $\mathcal{O}(n\log{}n)$, which makes $\partial\!\operatorname{Maj}$ more expensive than $\partial_{\neg}$, $\partial_{\wedge}$, $\partial_{\vee}$ and $\partial_{\Rightarrow}$ at training time. However, in the hard $\partial\mathbb{B}$ net we efficiently implement $\operatorname{Maj}$ as a discrete program that simply checks if the majority of bits are high. \subsection{Differentiable counting} @@ -285,12 +285,10 @@ \subsection{Differentiable counting} \begin{equation*} \begin{aligned} \operatorname{low-high}: [0,1]^{n} &\to [0,1]^{n+1},\\ -{\bf x} &\mapsto \left[ \partial\!\wedge\!(1, x_{1}), \partial\!\wedge\!(1 - x_{1}, x_{2}), \dots, \partial\!\wedge\!(1 - x_{n-1}, x_{n}), \partial\!\wedge\!(1-x_{n}, 1) \right]\text{.} +{\bf x} &\mapsto \left[ \partial_{\wedge}\!(1, x_{1}), \partial_{\wedge}\!(1 - x_{1}, x_{2}), \dots, \partial_{\wedge}\!(1 - x_{n-1}, x_{n}), \partial_{\wedge}\!(1-x_{n}, 1) \right]\text{.} \end{aligned} \end{equation*} -$\partial\!\operatorname{count-hot}({\bf x})$ outputs a 1-hot vector where the index of high bit is the number of low bits in ${\bf x}$. For example, $\partial\!\operatorname{count-hot}([0.1, 0.9, 0.2]) = [0.1, 0.2, \bold{0.8}, 0.1]\text{,}$ indicating that 2 bits are low, and $\partial\!\operatorname{count-hot}([0.6, 0.9, 0.7]) = [\bold{0.6}, 0.4, 0.3, 0.1]\text{,}$ indicating that 0 bits are low. - -Note that $\partial\!\operatorname{count-hot}$ is hard-equivalent to the boolean function +$\partial\!\operatorname{count-hot}({\bf x})$ outputs a 1-hot vector where the index of high bit is the number of low bits in ${\bf x}$. For example, $\partial\!\operatorname{count-hot}([0.1, 0.9, 0.2]) = [0.1, 0.2, \bold{0.8}, 0.1]\text{,}$ indicating that 2 bits are low, and $\partial\!\operatorname{count-hot}([0.6, 0.9, 0.7]) = [\bold{0.6}, 0.4, 0.3, 0.1]\text{,}$ indicating that 0 bits are low. Note that $\partial\!\operatorname{count-hot}$ is differentiable, gradient-rich and hard-equivalent to the boolean function \begin{equation*} \begin{aligned} \operatorname{count-hot}: \{0,1\}^{n} &\to \{0,1\}^{n+1}, \\ @@ -301,60 +299,59 @@ \subsection{Differentiable counting} \begin{equation*} \operatorname{k-of-n}({\bf x}, k) = \bigvee_{|S|=k} \bigwedge_{i\in S} x_i \bigwedge_{j\notin S} \neg x_j \end{equation*} -(see proposition \ref{prop:count}). However, in the hardened aspect of the $\partial\mathbb{B}$ net we efficiently implement $\operatorname{count-hot}$ as a discrete program that simply counts the number of low bits. +(see proposition \ref{prop:count}). However, in the hard $\partial\mathbb{B}$ net we efficiently implement $\operatorname{count-hot}$ as a discrete program that simply counts the number of low bits. We may then construct various kinds of boolean counting functions from $\partial\!\operatorname{count-hot}$. For example, $\partial\!\operatorname{count}({\bf x}, k)$ is straightforwardly $\partial\!\operatorname{count-hot}({\bf x})[k]$. -Note that $\partial\!\operatorname{count-hot}$ is differentiable and gradient-rich. We may construct various kinds of boolean counting functions from $\partial\!\operatorname{count-hot}$. For example, $\partial\!\operatorname{count}({\bf x}, k)$ is straightforwardly $\partial\!\operatorname{count-hot}({\bf x})[k]$. +This set of basic boolean functions is sufficient to learn non-trivial relationships from data. We now turn to composing these functions to form $\partial\mathbb{B}$ nets. -\subsection{Logical layers} +\subsection{$\partial\mathbb{B}$ net architectures} -Define +The possible variety of $\partial\mathbb{B}$ net architectures is probably similar to standard neural networks. Here we define basic layers sufficient to solve multiclass classification problems. Other kinds of layers, such as convolutional, or real encoders and decoders for solving regression problems, will be addressed in a sequel. + +A $\partial_{\neg} \!\operatorname{Layer}$ learns to negate a subset of elements of a vector: \begin{equation*} \begin{aligned} -\partial \neg \text{LAYER}: [0,1]^{n \times m} \times [0,1]^{m} &\to [0,1]^{n \times m}, \\ +\partial_{\neg} \!\operatorname{Layer}: [0,1]^{n \times m} \times [0,1]^{m} &\to [0,1]^{n \times m}, \\ ({\bf W}, {\bf x}) &\mapsto \begin{bmatrix} -\partial \neg(w_{1,1}, x_{1}) & \dots & \partial \neg(w_{1,m}, x_{m}) \\ +\partial_{\neg}(w_{1,1}, x_{1}) & \dots & \partial_{\neg}(w_{1,m}, x_{m}) \\ \vdots & \ddots & \vdots \\ -\partial \neg(w_{n,1}, x_{1}) & \dots & \partial \neg(w_{n,m}, x_{m}) +\partial_{\neg}(w_{n,1}, x_{1}) & \dots & \partial_{\neg}(w_{n,m}, x_{m}) \end{bmatrix} \end{aligned} \end{equation*} -where ${\bf W}$ is a matrix of weights and ${\bf x}$ is a vector of soft-bits. +where ${\bf W}$ is a weight matrix and $n$ is the width of the layer. -%[\partial \neg({\bf W}_{1}, {\bf x}), \dots, \partial \neg({\bf W}_{n}, {\bf x})] - -Define +A $\partial_{\wedge}\!\operatorname{Neuron}$ learns to logically $\wedge$ a subset of the elements of a vector: \begin{equation*} \begin{aligned} -\partial\!\wedge\!\text{NEURON}: [0,1]^{n} \times [0,1]^{n} &\to [0,1], \\ -({\bf w}, {\bf x}) &\mapsto \min(\partial \Rightarrow(w_{1}, x_{1}), \dots, \partial \Rightarrow(w_{n}, x_{n}))\text{,} +\partial_{\wedge}\!\operatorname{Neuron}: [0,1]^{n} \times [0,1]^{n} &\to [0,1], \\ +({\bf w}, {\bf x}) &\mapsto \min(\partial_{\Rightarrow}\!(w_{1}, x_{1}), \dots, \partial_{\Rightarrow}\!(w_{n}, x_{n}))\text{,} \end{aligned} \end{equation*} -where ${\bf w}$ is vector of weights and ${\bf x}$ is a vector of soft-bits. A single AND neuron maps $n$ soft-bit inputs to a single soft-bit value. A $\partial\!\wedge\!\text{LAYER}$ of $n$ neurons maps $m$ soft-bit inputs to $n$ soft-bit outputs. +where ${\bf w}$ is a weight vector. A $\partial_{\wedge}\!\operatorname{Neuron}$ maps $n$ inputs to a single soft-bit output. A $\partial_{\wedge}\!\operatorname{Layer}$ of $n$ neurons then maps $m$ inputs to $n$ outputs, where each output is the logical $\wedge$ of some subset of the input vector. In other words, a $\partial_{\wedge}\!\operatorname{Layer}$ of width $n$ can learn up to $n$ different conjunctions of subsets of its input. -Define +A $\partial_{\vee}\!\operatorname{Neuron}$ is defined similarly: \begin{equation*} \begin{aligned} -\partial\!\vee\!\text{NEURON}: [0,1]^{n} \times [0,1]^{n} &\to [0,1], \\ -({\bf w}, {\bf x}) &\mapsto \max(\partial \wedge(w_{1}, x_{1}), \dots, \partial \wedge(w_{n}, x_{n}))\text{.} +\partial_{\vee}\!\operatorname{Neuron}: [0,1]^{n} \times [0,1]^{n} &\to [0,1], \\ +({\bf w}, {\bf x}) &\mapsto \max(\partial_{\wedge}\!(w_{1}, x_{1}), \dots, \partial_{\wedge}\!(w_{n}, x_{n}))\text{.} \end{aligned} \end{equation*} -A single $\vee$ neuron maps $n$ soft-bit inputs to a single soft-bit value. A $\partial\!\vee\!\text{LAYER}$ of $n$ neurons maps $m$ soft-bit inputs to $n$ soft-bit outputs. +A $\partial_{\vee}\!\operatorname{Layer}$ of width $n$ can learn up to $n$ different disjunctions of subsets of its input. -\subsection{Architectures} +We can compose $\partial_{\neg}$, $\partial_{\wedge}$ and $\partial_{\vee}$ layers to learn boolean formulae of arbitrary width and depth. -todo: architectures +In classification problems the final layer of a standard neural network is typically interpreted as a vector of real-valued logits, one for each possible label. The index of the maximum logit indicates the most probable label. However, if we interpret the final layer of a $\partial\mathbb{B}$ net as a vector of logits then we violate hard-equivalence. Instead, for classification problems, the final layer of a $\partial\mathbb{B}$ net is a 1-hot soft-bit vector, constructed using $\partial\!\operatorname{count-hot}$, where the index of the high bit indicates the predicted label. The soft net outputs a 1-hot soft-bit vector, and the hard net outputs a 1-hot boolean vector. -todo: multi-label classification Define \begin{equation*} \begin{aligned} -\partial\text{HARDEN}: [0,1]^{n} &\to [0,1]^{n}, \\ +\partial\!\operatorname{Harden}: [0,1]^{n} &\to [0,1]^{n}, \\ {\bf x} &\mapsto \operatorname{harden}({\bf x})\text{.} \end{aligned} \end{equation*} -A $\partial\text{HARDEN}$ layer maps $m$ soft-bit inputs to $n$ hard-bit outputs. The $\operatorname{harden}$ function is not differentiable and therefore we use the straight-through estimator \citep{DBLP:journals/corr/BengioLC13} during backpropagation. +A $\partial\!\operatorname{Harden}$ layer maps $m$ soft-bit inputs to $n$ hard-bit outputs. The $\operatorname{harden}$ function is not differentiable and therefore we use the straight-through estimator \citep{DBLP:journals/corr/BengioLC13} during backpropagation. \section{Experiments} @@ -370,7 +367,7 @@ \subsection{Noisy XOR} \begin{figure}[t] \centering \includegraphics[width=1.0\textwidth]{noisy-xor-architecture.png} - \caption{{\em A $\partial\mathbb{B}$ net for the noisy-xor problem}. The net concatenates the soft-bit input, ${\bf x}$ (length 12), with its negation, ${\bf 1 - x}$, and supplies the resulting vector (length 24) to a $\partial\!\wedge\!\text{LAYER}$ (width 32), $\partial\!\vee\!\text{LAYER}$ (width 32), $\partial \neg \text{LAYER}$ (width 16), and a final $\partial\!\operatorname{Maj}$ to produce a single soft-bit $y \in [0,1]$ (to predict odd parity) and its negation $1-y$ (to predict even parity). The net's weights, once hardened, consume $288$ bytes.} + \caption{{\em A $\partial\mathbb{B}$ net for the noisy-xor problem}. The net concatenates the soft-bit input, ${\bf x}$ (length 12), with its negation, ${\bf 1 - x}$, and supplies the resulting vector (length 24) to a $\partial_{\wedge}\!\!\operatorname{Layer}$ (width 32), $\partial_{\vee}\!\!\operatorname{Layer}$ (width 32), $\partial_{\neg} \!\operatorname{Layer}$ (width 16), and a final $\partial\!\operatorname{Maj}$ to produce a single soft-bit $y \in [0,1]$ (to predict odd parity) and its negation $1-y$ (to predict even parity). The net's weights, once hardened, consume $288$ bytes.} \label{fig:noisy-xor-architecture} \end{figure} @@ -418,13 +415,13 @@ \section*{Appendix} \section{Proofs} \begin{prop}\label{prop:not} - $\partial \neg(x,y) \equiv \neg (x \oplus y)$. + $\partial_{\neg}(x,y) \equiv \neg (x \oplus y)$. \begin{proof} Table \ref{not-table} is the truth table of the boolean function $\neg (x \oplus w)$, where $h(x) = \operatorname{harden}(x)$. \begin{table} \begin{center} \begin{tabular}{cccccc} - \multicolumn{1}{c}{$x$} &\multicolumn{1}{c}{$y$} &\multicolumn{1}{c}{$h(x)$} &\multicolumn{1}{c}{$h(y)$} &\multicolumn{1}{c}{$\partial \neg(h(x), h(y))$} &\multicolumn{1}{c}{$h(\partial \neg(h(x), h(y)))$} + \multicolumn{1}{c}{$x$} &\multicolumn{1}{c}{$y$} &\multicolumn{1}{c}{$h(x)$} &\multicolumn{1}{c}{$h(y)$} &\multicolumn{1}{c}{$\partial_{\neg}(h(x), h(y))$} &\multicolumn{1}{c}{$h(\partial_{\neg}(h(x), h(y)))$} \\ \hline \\ $\left[0, \frac{1}{2}\right]$ & $\left[0, \frac{1}{2}\right]$ & 0 & 0 & 1 & 1\\[0.1cm] $\left(\frac{1}{2}, 1\right]$ & $\left[0, \frac{1}{2}\right]$ &1 & 0 & 0 & 0\\[0.1cm] @@ -432,7 +429,7 @@ \section{Proofs} $\left(\frac{1}{2}, 1\right]$ & $\left(\frac{1}{2}, 1\right]$ &1 & 1 & 1 & 1\\[0.1cm] \end{tabular} \end{center} - \caption{$\partial \neg(x,y) \equiv \neg (y \oplus x)$.}\label{not-table} + \caption{$\partial_{\neg}(x,y) \equiv \neg (y \oplus x)$.}\label{not-table} \end{table} \end{proof} \end{prop} @@ -446,13 +443,13 @@ \section{Proofs} \begin{prop}\label{prop:and} - $\partial\!\wedge\!(x,y) \equiv x \wedge y$. + $\partial_{\wedge}\!(x,y) \equiv x \wedge y$. \begin{proof} Table \ref{and-table} is the truth table of the boolean function $x \wedge y$, where $h(x) = \operatorname{harden}(x)$.. \begin{table} \begin{center} \begin{tabular}{cccccc} - \multicolumn{1}{c}{$x$} &\multicolumn{1}{c}{$y$} &\multicolumn{1}{c}{$h(x)$} &\multicolumn{1}{c}{$h(y)$} &\multicolumn{1}{c}{$\partial \wedge(h(x), h(y))$} &\multicolumn{1}{c}{$h(\partial \wedge(h(x), h(y)))$} + \multicolumn{1}{c}{$x$} &\multicolumn{1}{c}{$y$} &\multicolumn{1}{c}{$h(x)$} &\multicolumn{1}{c}{$h(y)$} &\multicolumn{1}{c}{$\partial_{\wedge}(h(x), h(y))$} &\multicolumn{1}{c}{$h(\partial_{\wedge}(h(x), h(y)))$} \\ \hline \\ $\left[0, \frac{1}{2}\right]$ & $\left[0, \frac{1}{2}\right]$ & 0 & 0 & 0 & 0\\[0.1cm] $\left(\frac{1}{2}, 1\right]$ & $\left[0, \frac{1}{2}\right]$ &1 & 0 & $\frac{1}{4}$ & 0\\[0.1cm] @@ -460,19 +457,19 @@ \section{Proofs} $\left(\frac{1}{2}, 1\right]$ & $\left(\frac{1}{2}, 1\right]$ &1 & 1 & 1 & 1\\[0.1cm] \end{tabular} \end{center} - \caption{$\partial \wedge(x,y) \equiv x \wedge y$.}\label{and-table} + \caption{$\partial_{\wedge}(x,y) \equiv x \wedge y$.}\label{and-table} \end{table} \end{proof} \end{prop} \begin{prop}\label{prop:or} - $\partial\!\vee\!(x,y) \equiv x \vee y$. + $\partial_{\vee}\!(x,y) \equiv x \vee y$. \begin{proof} Table \ref{or-table} is the truth table of the boolean function $x \vee y$, where $h(x) = \operatorname{harden}(x)$.. \begin{table} \begin{center} \begin{tabular}{cccccc} - \multicolumn{1}{c}{$x$} &\multicolumn{1}{c}{$y$} &\multicolumn{1}{c}{$h(x)$} &\multicolumn{1}{c}{$h(y)$} &\multicolumn{1}{c}{$\partial \vee(h(x), h(y))$} &\multicolumn{1}{c}{$h(\partial \vee(h(x), h(y)))$} + \multicolumn{1}{c}{$x$} &\multicolumn{1}{c}{$y$} &\multicolumn{1}{c}{$h(x)$} &\multicolumn{1}{c}{$h(y)$} &\multicolumn{1}{c}{$\partial_{\vee}(h(x), h(y))$} &\multicolumn{1}{c}{$h(\partial_{\vee}(h(x), h(y)))$} \\ \hline \\ $\left[0, \frac{1}{2}\right]$ & $\left[0, \frac{1}{2}\right]$ & 0 & 0 & 0 & 0\\[0.1cm] $\left(\frac{1}{2}, 1\right]$ & $\left[0, \frac{1}{2}\right]$ &1 & 0 & $\frac{3}{4}$ & 1\\[0.1cm] @@ -480,13 +477,13 @@ \section{Proofs} $\left(\frac{1}{2}, 1\right]$ & $\left(\frac{1}{2}, 1\right]$ &1 & 1 & 1 & 1\\[0.1cm] \end{tabular} \end{center} - \caption{$\partial \vee(x,y) \equiv x \vee y$.}\label{or-table} + \caption{$\partial_{\vee}(x,y) \equiv x \vee y$.}\label{or-table} \end{table} \end{proof} \end{prop} \begin{prop}\label{prop:implies} - $\partial\!\Rightarrow\!(x,y) \equiv x \Rightarrow y$. + $\partial_{\Rightarrow}\!(x,y) \equiv x \Rightarrow y$. \begin{proof} Table \ref{implies-table} is the truth table of the boolean function $x \Rightarrow y$, where $h(x) = \operatorname{harden}(x)$.. \begin{table} @@ -541,7 +538,7 @@ \section{Proofs} \begin{equation*} \begin{aligned} \partial\text{AND-LAYER}: [0,1]^{n \times m} \times [0,1]^{m} &\to [0,1]^{n}, \\ -({\bf W}, {\bf x}) &\mapsto [\partial\wedge\text{NEURON}({\bf W}_{1}, {\bf x}), \dots, \partial\wedge\text{NEURON}({\bf W}_{n}, {\bf x})] +({\bf W}, {\bf x}) &\mapsto [\partial_{\wedge}\operatorname{Neuron}({\bf W}_{1}, {\bf x}), \dots, \partial_{\wedge}\operatorname{Neuron}({\bf W}_{n}, {\bf x})] \end{aligned} \end{equation*} \end{comment} From 62ec548b9932fb04a6331b5f8b00bad1ef8196d7 Mon Sep 17 00:00:00 2001 From: Ian Wright Date: Thu, 23 Mar 2023 11:12:23 +0000 Subject: [PATCH 052/113] more --- docs/binary-iris-architecture.png | 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For example, $\partial\!\operatorname{Maj}({\bf x})$ is a boolean counting function where $c({\bf x}) := |\{x_{i} : x_{i} = 1 \}| \geq \lceil \frac{n}{2} \rceil$. We aim to construct a differentiable analogue of $\operatorname{count}({\bf x}, k)$ where $c({\bf x}) := |\{x_{i} : x_{i} = 1 \}| = k$ (i.e. `exactly $k$ high'). This will be useful for multiclass classification problems where we interpret $k$ as a class prediction. +A boolean counting function $f({\bf x})$ has the value $1$ if a counting predicate, $c({\bf x})$, holds over its $n$ inputs. For example, $\partial\!\operatorname{Maj}({\bf x})$ is a boolean counting function where $c({\bf x}) := |\{x_{i} : x_{i} = 1 \}| \geq \lceil \frac{n}{2} \rceil$. We aim to construct a differentiable analogue of $\operatorname{count}({\bf x}, k)$ where $c({\bf x}) := |\{x_{i} : x_{i} = 1 \}| = k$ (i.e. `exactly $k$ high'). This is useful for multiclass classification problems where we interpret $k$ as a class prediction. As before, we use $\operatorname{sort}$ to trade-off time for memory costs. Observe that if the elements of ${\bf x}$ are in ascending order then, if any soft-bits are high, there exists a unique contiguous pair of indices $(i,i+1)$ where $x_{i}$ is low and $x_{i+1}$ is high, where index $i$ is a direct count of the number of soft-bits that are low in ${\bf x}$. In consequence, define \begin{equation*} @@ -299,15 +297,15 @@ \subsection{Differentiable counting} \begin{equation*} \operatorname{k-of-n}({\bf x}, k) = \bigvee_{|S|=k} \bigwedge_{i\in S} x_i \bigwedge_{j\notin S} \neg x_j \end{equation*} -(see proposition \ref{prop:count}). However, in the hard $\partial\mathbb{B}$ net we efficiently implement $\operatorname{count-hot}$ as a discrete program that simply counts the number of low bits. We may then construct various kinds of boolean counting functions from $\partial\!\operatorname{count-hot}$. For example, $\partial\!\operatorname{count}({\bf x}, k)$ is straightforwardly $\partial\!\operatorname{count-hot}({\bf x})[k]$. +(see proposition \ref{prop:count}). However, in the hard $\partial\mathbb{B}$ net we efficiently implement $\operatorname{count-hot}$ as a discrete program that simply counts the number of low bits. We can construct various kinds of boolean counting functions from $\partial\!\operatorname{count-hot}$. For example, $\partial\!\operatorname{count}({\bf x}, k)$ is straightforwardly $\partial\!\operatorname{count-hot}({\bf x})[k]$ where we can again use margin-packing to ensure that this single soft-bit is gradient-rich. -This set of basic boolean functions is sufficient to learn non-trivial relationships from data. We now turn to composing these functions to form $\partial\mathbb{B}$ nets. +This set of basic boolean functions is sufficient to learn non-trivial relationships from data. We now turn to constructing $\partial\mathbb{B}$ nets from compositions of these basic functions. -\subsection{$\partial\mathbb{B}$ net architectures} +\subsection{Boolean logic layers} -The possible variety of $\partial\mathbb{B}$ net architectures is probably similar to standard neural networks. Here we define basic layers sufficient to solve multiclass classification problems. Other kinds of layers, such as convolutional, or real encoders and decoders for solving regression problems, will be addressed in a sequel. +The possible variety of $\partial\mathbb{B}$ net architectures is similar to standard neural networks. Here we merely define some basic layers that are sufficient for classification problems. Other kinds of layers, such as convolutional, or real encoders and decoders for regression problems, will be addressed in a sequel. -A $\partial_{\neg} \!\operatorname{Layer}$ learns to negate a subset of elements of a vector: +A $\partial_{\neg} \!\operatorname{Layer}$ of width $n$ learns to negate up to $n$ different subsets of the elements of its input vector: \begin{equation*} \begin{aligned} \partial_{\neg} \!\operatorname{Layer}: [0,1]^{n \times m} \times [0,1]^{m} &\to [0,1]^{n \times m}, \\ @@ -319,9 +317,9 @@ \subsection{$\partial\mathbb{B}$ net architectures} \end{bmatrix} \end{aligned} \end{equation*} -where ${\bf W}$ is a weight matrix and $n$ is the width of the layer. +where ${\bf x}$ is a soft-bit input vector, ${\bf W}$ is a weight matrix and $n$ is the layer width. -A $\partial_{\wedge}\!\operatorname{Neuron}$ learns to logically $\wedge$ a subset of the elements of a vector: +A $\partial_{\wedge}\!\operatorname{Neuron}$ learns to logically $\wedge$ a subset of the elements of its input vector: \begin{equation*} \begin{aligned} \partial_{\wedge}\!\operatorname{Neuron}: [0,1]^{n} \times [0,1]^{n} &\to [0,1], \\ @@ -337,21 +335,22 @@ \subsection{$\partial\mathbb{B}$ net architectures} ({\bf w}, {\bf x}) &\mapsto \max(\partial_{\wedge}\!(w_{1}, x_{1}), \dots, \partial_{\wedge}\!(w_{n}, x_{n}))\text{.} \end{aligned} \end{equation*} -A $\partial_{\vee}\!\operatorname{Layer}$ of width $n$ can learn up to $n$ different disjunctions of subsets of its input. +In consequence, a $\partial_{\vee}\!\operatorname{Layer}$ of width $n$ can learn up to $n$ different disjunctions of subsets of its input. -We can compose $\partial_{\neg}$, $\partial_{\wedge}$ and $\partial_{\vee}$ layers to learn boolean formulae of arbitrary width and depth. +We compose $\partial_{\neg}$, $\partial_{\wedge}$ and $\partial_{\vee}$ layers to learn boolean formulae of arbitrary width and depth. -In classification problems the final layer of a standard neural network is typically interpreted as a vector of real-valued logits, one for each possible label. The index of the maximum logit indicates the most probable label. However, if we interpret the final layer of a $\partial\mathbb{B}$ net as a vector of logits then we violate hard-equivalence. Instead, for classification problems, the final layer of a $\partial\mathbb{B}$ net is a 1-hot soft-bit vector, constructed using $\partial\!\operatorname{count-hot}$, where the index of the high bit indicates the predicted label. The soft net outputs a 1-hot soft-bit vector, and the hard net outputs a 1-hot boolean vector. +\subsection{Classification layers} +In classification problems the final layer of a neural network is typically interpreted as a vector of real-valued logits, one for each label. The index of the maximum logit indicates the most probable label. However, we cannot interpret the final layer of a $\partial\mathbb{B}$ net as a vector of logits without violating hard-equivalence. Instead, for classification problems, the final layer of a $\partial\mathbb{B}$ net is a 1-hot soft-bit vector, constructed using $\partial\!\operatorname{count-hot}$, where the index of the high bit indicates the predicted label. The hard net then outputs a 1-hot boolean vector with the identical interpretation. -Define +In addition, at training time, loss functions should be a function of hardened bits, otherwise gradient descent may non-optimally traverse trajectories that take no account of the hard threshold at $1/2$. For example, say that an example is correctly classified by a 1-hot vector with high bit $x=0.51$. Updating the net's weights to change this value to $x=0.52$ won't increase classification accuracy but may prevent updating the weights to correctly classify another example in the training data. For this reason, $\partial\mathbb{B}$ nets have a final `hardening' layer to ensure that loss is a function of hard, not soft, bits: \begin{equation*} \begin{aligned} \partial\!\operatorname{Harden}: [0,1]^{n} &\to [0,1]^{n}, \\ {\bf x} &\mapsto \operatorname{harden}({\bf x})\text{.} \end{aligned} \end{equation*} -A $\partial\!\operatorname{Harden}$ layer maps $m$ soft-bit inputs to $n$ hard-bit outputs. The $\operatorname{harden}$ function is not differentiable and therefore we use the straight-through estimator \citep{DBLP:journals/corr/BengioLC13} during backpropagation. +The $\operatorname{harden}$ function is not differentiable and therefore we use the straight-through estimator \citep{DBLP:journals/corr/BengioLC13} during backpropagation. However, by restricting the straight-through estimator to the very final layer we avoid compounding gradient errors to deeper parts of the network. \section{Experiments} @@ -360,9 +359,46 @@ \subsection{Hardening} \subsection{Binary Iris} -\subsection{Noisy XOR} +\begin{figure}[t] + \centering + \includegraphics[width=1.0\textwidth]{binary-iris-architecture.png} + \caption{{\em A $\partial\mathbb{B}$ net for the binary iris problem}.} + \label{fig:binary-iris-architecture} +\end{figure} -The noisy XOR dataset \citep{noisy-xor-dataset} is an adversarial parity problem with noisy non-informative features. The dataset consists of 10K examples with 12 boolean inputs and a target label (where 0 = odd and 1 = even) that is a XOR function of 2 inputs. The remaining 10 inputs are entirely random. We train on 50\% of the data where, additionally, 40\% of the labels are inverted. + +\begin{comment} +""" +| Technique/Accuracy | Mean | 5 %ile | 95 %ile | Min | Max | +| ------------------ | -------------- | ------- | ------- | ------ | ------ | +| Tsetlin | 95.0 +/- 0.2 | 86.7 | 100.0 | 80.0 | 100.0 | +| dB | 93.9 +/- 0.1 | 86.7 | 100.0 | 80.0 | 100.0 | +| Neural network | 93.8 +/- 0.2 | 86.7 | 100.0 | 80.0 | 100.0 | +| SVM | 93.6 +/- 0.3 | 86.7 | 100.0 | 76.7 | 100.0 | +| Naive Bayes | 91.6 +/- 0.3 | 83.3 | 96.7 | 70.0 | 100.0 | + +Source: https://arxiv.org/pdf/1804.01508.pdf +""" +\end{comment} + +\begin{table}[t] + \centering + \begin{tabular}{llllll} + \cline{2-6} + \multicolumn{1}{c}{} & \multicolumn{5}{c}{\textbf{accuracy}} \\ \cline{2-6} + \multicolumn{1}{l|}{} & \multicolumn{1}{l|}{mean} & \multicolumn{1}{l|}{5 \%ile} & \multicolumn{1}{l|}{95 \%ile} & \multicolumn{1}{l|}{min} & \multicolumn{1}{l|}{max} \\ \hline + \multicolumn{1}{|l|}{Tsetlin} & \multicolumn{1}{l|}{95.0 +/- 0.2} & \multicolumn{1}{l|}{86.7} & \multicolumn{1}{l|}{100.0} & \multicolumn{1}{l|}{80.0} & \multicolumn{1}{l|}{100.0} \\ \hline + \multicolumn{1}{|l|}{$\partial\mathbb{B}$} & \multicolumn{1}{l|}{\textbf{93.9 +/- 0.1}} & \multicolumn{1}{l|}{\textbf{86.7}} & \multicolumn{1}{l|}{\textbf{100.0}} & \multicolumn{1}{l|}{\textbf{80.0}} & \multicolumn{1}{l|}{\textbf{100.0}} \\ \hline + \multicolumn{1}{|l|}{neural network} & \multicolumn{1}{l|}{93.8 +/- 0.2} & \multicolumn{1}{l|}{86.7} & \multicolumn{1}{l|}{100.0} & \multicolumn{1}{l|}{80.0} & \multicolumn{1}{l|}{100.0} \\ \hline + \multicolumn{1}{|l|}{SVM} & \multicolumn{1}{l|}{93.6 +/- 0.3} & \multicolumn{1}{l|}{86.7} & \multicolumn{1}{l|}{100.0} & \multicolumn{1}{l|}{76.7} & \multicolumn{1}{l|}{100.0} \\ \hline + \multicolumn{1}{|l|}{naive Bayes} & \multicolumn{1}{l|}{91.6 +/- 0.3} & \multicolumn{1}{l|}{83.3} & \multicolumn{1}{l|}{96.7} & \multicolumn{1}{l|}{70.0} & \multicolumn{1}{l|}{100.0} \\ \hline + \end{tabular} + \caption{{\em Ranked binary iris results}} + \label{tab:binary-iris-results} +\end{table} + + +\subsection{Noisy XOR} \begin{figure}[t] \centering @@ -371,9 +407,11 @@ \subsection{Noisy XOR} \label{fig:noisy-xor-architecture} \end{figure} +The noisy XOR dataset \citep{noisy-xor-dataset} is an adversarial parity problem with noisy non-informative features. The dataset consists of 10K examples with 12 boolean inputs and a target label (where 0 = odd and 1 = even) that is a XOR function of 2 inputs. The remaining 10 inputs are entirely random. We train on 50\% of the data where, additionally, 40\% of the labels are inverted. + We initialized the network described in figure \ref{fig:noisy-xor-architecture} using the XX policy and then trained for 2000 epochs with the RAdam optimizer \citep{Liu2020On} and softmax cross-entropy loss. We measure the accuracy of the final net on the test data (to avoid handpicking the best configuration). Table \ref{tab:noisy-xor-results} compares the $\partial\mathbb{B}$ net against other classifiers \citep{granmo18} and reports the mean accuracy with 95\% confidence intervals obtained over 100 replications of the experiment with different random seeds. -\begin{table}[h] +\begin{table}[t] \centering \begin{tabular}{llllll} \cline{2-6} diff --git a/docs/noisy-xor-architecture.png b/docs/noisy-xor-architecture.png index 257ecf87beec1176fe75eb75ff24d83d0d3ef45c..c7afd158eb9f1ac987a6c01b751fea6c9245906e 100644 GIT binary patch literal 32025 zcmeFZWmJ@F-#)C{t!x8P5D-vMDG>ps88P$`k2I|ih?VTggD zbAS5YAG@uGLY%Lh-|K1V%no4m2@ z?nx%@SI)G*^mdr4b$aJI#+*JcddNOnnV&~#wlrX`s5*e-*Tr6%CZ&*XX zF`F#&M>bjpfmt+jZ%(+%=;}@c?=0narge(q+KO>=osqY4p?I@s5vLU$6&-i{9EV2H zE;0*q+A$uraL@MbGCT~ohJ!6s*rW;U9N>}(R}YFZ}}OG_}Y+RTB5*ax#Q*Xo19rXYxivsP=Sk9c3{_}bI#C<3yURow4@BloX?AyhW{L zR?#?F`ax4?r%umYP8d3VA5F&YKylHD!DHf>&~($Y2=Hc3b`Epyuhy}QV2r;pR+W@& zmR7p&z=9>Y2ciURCU2lN_w_J?n$%QJi`rXTcZ@p4%PaWy5hVIflEr0cz6r^pik*h4 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8 ++--- neurallogic/hard_dropout.py | 7 +++-- neurallogic/hard_or.py | 28 +++++++++++------ tests/test_iris.py | 62 +++++++++++++++++++++++-------------- 5 files changed, 86 insertions(+), 47 deletions(-) diff --git a/neurallogic/hard_and.py b/neurallogic/hard_and.py index e57113d..2dd7259 100644 --- a/neurallogic/hard_and.py +++ b/neurallogic/hard_and.py @@ -6,12 +6,6 @@ from neurallogic import hard_masks, neural_logic_net, symbolic_generation, initialization -# TODO: seperate and operation from mask operation -def soft_and_neuron(w, x): - x = jax.vmap(hard_masks.soft_mask_to_true_margin, 0, 0)(w, x) - return jax.numpy.min(x) - - def soft_and(x, y): m = jax.numpy.minimum(x, y) return jax.numpy.where( @@ -20,10 +14,26 @@ def soft_and(x, y): m + 0.5 * (x + y) * (0.5 - m), ) -# This doesn't work well +def soft_and_vec(x): + m = jax.numpy.min(x) + mean = jax.numpy.mean(x) + delta = jax.numpy.abs(mean - 0.5) + return jax.numpy.where( + 2 * m > 1, + 0.5 + delta, + m + delta + ) + +# TODO: seperate and operation from mask operation +def soft_and_neuron(w, x): + x = jax.vmap(hard_masks.soft_mask_to_true_margin, 0, 0)(w, x) + #x = jax.vmap(hard_masks.soft_mask_to_true, 0, 0)(w, x) + return jax.numpy.min(x) + +# TODO: doesn't seem to work as well def soft_and_neuron_deprecated(w, x): - x = jax.vmap(hard_masks.soft_mask_to_true, 0, 0)(w, x) - return jax.lax.reduce(x, 1.0, soft_and, [0]) + x = jax.vmap(hard_masks.soft_mask_to_true_margin, 0, 0)(w, x) + return soft_and_vec(x) def hard_and_neuron(w, x): x = jax.vmap(hard_masks.hard_mask_to_true, 0, 0)(w, x) diff --git a/neurallogic/hard_count.py b/neurallogic/hard_count.py index 27e72d4..4ab011a 100644 --- a/neurallogic/hard_count.py +++ b/neurallogic/hard_count.py @@ -1,11 +1,11 @@ import jax from flax import linen as nn -from neurallogic import neural_logic_net, symbolic_generation +from neurallogic import neural_logic_net, symbolic_generation, hard_and -def high_to_low(x, y): - return jax.numpy.minimum(1 - x, y) +def low_to_high(x, y): + return hard_and.soft_and(1 - x, y) def soft_count(x: jax.numpy.array): """ @@ -32,7 +32,7 @@ def soft_count(x: jax.numpy.array): low = jax.numpy.array([0.0]) high = jax.numpy.array([1.0]) sorted_x = jax.numpy.concatenate([low, sorted_x, high]) - return jax.vmap(high_to_low)(sorted_x[:-1], sorted_x[1:]) + return jax.vmap(low_to_high)(sorted_x[:-1], sorted_x[1:]) def hard_count(x: jax.numpy.array): # We simply count the number of low bits diff --git a/neurallogic/hard_dropout.py b/neurallogic/hard_dropout.py index 2f92cae..3c20ccf 100644 --- a/neurallogic/hard_dropout.py +++ b/neurallogic/hard_dropout.py @@ -1,4 +1,4 @@ -from typing import Optional, Sequence +from typing import Optional, Sequence, Callable import jax from flax import linen as nn @@ -30,8 +30,8 @@ class SoftHardDropout(nn.Module): broadcast_dims: Sequence[int] = () deterministic: Optional[bool] = None rng_collection: str = "dropout" - dropout_value: float = 0.0 dtype: jax.numpy.dtype = jax.numpy.float32 + dropout_function: Callable = lambda x: jax.numpy.full_like(x, 0.0) @nn.compact def __call__(self, inputs, deterministic: Optional[bool] = None): @@ -65,9 +65,12 @@ def __call__(self, inputs, deterministic: Optional[bool] = None): broadcast_shape[dim] = 1 mask = random.bernoulli(rng, p=keep_prob, shape=broadcast_shape) mask = jax.numpy.broadcast_to(mask, inputs.shape) + """ masked_values = jax.numpy.full_like( inputs, self.dropout_value, dtype=self.dtype ) + """ + masked_values = jax.vmap(self.dropout_function)(inputs) return lax.select(mask, inputs, masked_values) diff --git a/neurallogic/hard_or.py b/neurallogic/hard_or.py index 417066a..7aefdb0 100644 --- a/neurallogic/hard_or.py +++ b/neurallogic/hard_or.py @@ -11,12 +11,6 @@ ) -# TODO: seperate out the or operation from the mask operation -def soft_or_neuron(w, x): - x = jax.vmap(hard_masks.soft_mask_to_false_margin, 0, 0)(w, x) - return jax.numpy.max(x) - - def soft_or(x, y): m = jax.numpy.maximum(x, y) return jax.numpy.where( @@ -25,10 +19,26 @@ def soft_or(x, y): m + 0.5 * (x + y) * (0.5 - m), ) -# This doesn't work well +def soft_or_vec(x): + m = jax.numpy.max(x) + mean = jax.numpy.mean(x) + delta = jax.numpy.abs(mean - 0.5) + return jax.numpy.where( + 2 * m > 1, + 0.5 + delta, + m + delta + ) + + +# TODO: seperate out the or operation from the mask operation +def soft_or_neuron(w, x): + x = jax.vmap(hard_masks.soft_mask_to_false_margin, 0, 0)(w, x) + return jax.numpy.max(x) + +# TODO: doesn't seem to work as well def soft_or_neuron_deprecated(w, x): - x = jax.vmap(hard_masks.soft_mask_to_true, 0, 0)(w, x) - return jax.lax.reduce(x, 0.0, soft_or, [0]) + x = jax.vmap(hard_masks.soft_mask_to_false_margin, 0, 0)(w, x) + return soft_or_vec(x) def hard_or_neuron(w, x): diff --git a/tests/test_iris.py b/tests/test_iris.py index afcb7ab..5193abe 100644 --- a/tests/test_iris.py +++ b/tests/test_iris.py @@ -145,10 +145,6 @@ def nln_iris(type, x, training: bool): Source: https://arxiv.org/pdf/1804.01508.pdf """ - -# TODO: implement count layer, k-high neuron, and multi-label classification -# to avoid the need for the harden layer - # Using majority without margin # mean: 94.18, sem: 0.13, min: 80.00, max: 100.00, 5%: 86.67, 95%: 100.00 # Using majority with margin @@ -169,27 +165,57 @@ def nln_binary_iris_1(type, x, training: bool): x = x.sum(-1) return x +""" +| Technique/Accuracy | Mean | 5 %ile | 95 %ile | Min | Max | +| ------------------ | -------------- | ------- | ------- | ------ | ------ | +| Tsetlin | 95.0 +/- 0.2 | 86.7 | 100.0 | 80.0 | 100.0 | +| dB | 93.9 +/- 0.1 | 86.7 | 100.0 | 80.0 | 100.0 | +| Neural network | 93.8 +/- 0.2 | 86.7 | 100.0 | 80.0 | 100.0 | +| SVM | 93.6 +/- 0.3 | 86.7 | 100.0 | 76.7 | 100.0 | +| Naive Bayes | 91.6 +/- 0.3 | 83.3 | 96.7 | 70.0 | 100.0 | + +Source: https://arxiv.org/pdf/1804.01508.pdf +""" +# mean: 93.88, sem: 0.12, min: 80.00, max: 100.00, 5%: 86.67, 95%: 100.00 def nln_binary_iris(type, x, training: bool): dtype = jax.numpy.float64 y = hard_vmap.vmap(type)((lambda x: 1 - x, lambda x: 1 - x, lambda x: symbolic_primitives.symbolic_not(x)))(x) x = hard_concatenate.concatenate(type)([x, y], 0) - layer_size = 16 + layer_size = 59 x = hard_and.and_layer(type)( layer_size, dtype=dtype, - weights_init=initialization.initialize_bernoulli(0.01, 0.3, 0.501), + weights_init=initialization.initialize_bernoulli(0.0, 0.3, 0.501), + )(x) + x = hard_dropout.hard_dropout(type)( + rate=0.05, + dropout_function=lambda x: 1-x, + deterministic=not training, + dtype=dtype, )(x) - x = x.ravel() - x = x.reshape((num_classes - 1, int(x.shape[0] / (num_classes - 1)))) - x = hard_majority.majority_layer(type)()(x) ######################################################## x = jax.numpy.array([x]) # TODO: shouldn't need to do this - x = hard_count.count_layer(type)()(x) - x = x.ravel() + # count the number of high bits to yield layer_size+1 outputs + x = hard_count.count_layer(type)()(x) + # split into num_classes equally sized bit buckets + x = x.ravel() # TODO: shouldn't need to do this x = x.reshape((num_classes, int(x.shape[0] / num_classes))) + # take the logical or of each bucket + # TODO: create a specialised layer for this + x = hard_vmap.vmap(type)(( + lambda x: jax.numpy.max(x), + # This is conceptually wrong + #lambda x: hard_or.soft_or_vec(x), # I don't want the other bits in the bucket to be high (if correct label) + lambda x: jax.numpy.max(x), + lambda x: symbolic_primitives.symbolic_reduce_or(x)))(x) + x = x.ravel() + x = harden_layer.harden_layer(type)(x) + x = x.reshape((num_classes, int(x.shape[0] / num_classes))) # TODO: shouldn't need to do this x = x.sum(-1) return x + + def batch_nln_iris(type, x, training: bool): return jax.vmap(lambda x: nln_iris(type, x, training))(x) @@ -205,7 +231,6 @@ class TrainState(train_state.TrainState): def create_train_state(net, rng, dropout_rng, config): mock_input = jax.numpy.ones([1, num_features]) soft_weights = net.init(rng, mock_input, training=False)["params"] - #tx = optax.sgd(config.learning_rate, config.momentum) tx = optax.radam(learning_rate=config.learning_rate) return TrainState.create( apply_fn=net.apply, params=soft_weights, tx=tx, dropout_rng=dropout_rng @@ -216,13 +241,6 @@ def create_train_state(net, rng, dropout_rng, config): def update_model(state, grads): return state.apply_gradients(grads=grads) -def my_loss(predictions, targets): - return jax.vmap(lambda x: jax.numpy.where(x < 0.45, 0.0, x*x))(predictions - targets) - -def my_loss(predictions, targets): - x = predictions - targets - return x*x - def apply_model_with_grad_impl(state, features, labels, dropout_rng, training: bool): dropout_train_rng = jax.random.fold_in(key=dropout_rng, data=state.step) @@ -236,8 +254,6 @@ def loss_fn(params): one_hot = jax.nn.one_hot(labels, num_classes) loss = jax.numpy.mean( optax.softmax_cross_entropy(logits=logits, labels=one_hot) - #optax.l2_loss(logits, one_hot) - #my_loss(logits, one_hot) ) return loss, logits @@ -344,8 +360,8 @@ def get_config(): config = ml_collections.ConfigDict() config.learning_rate = 0.01 # sgd = 0.1 config.momentum = 0.9 - config.batch_size = 120 - config.num_epochs = 4000 # 20000 # 500 for paper + config.batch_size = 60 + config.num_epochs = 1000 return config From b89246fad1f06b94b4b6b2432749f43355406517 Mon Sep 17 00:00:00 2001 From: Ian Wright Date: Thu, 23 Mar 2023 15:08:54 +0000 Subject: [PATCH 054/113] more --- docs/binary-iris-architecture.png | Bin 42719 -> 43325 bytes docs/db-net.png | Bin 34540 -> 34785 bytes docs/db.tex | 48 ++++++++++++++++-------------- 3 files changed, 26 insertions(+), 22 deletions(-) diff --git a/docs/binary-iris-architecture.png b/docs/binary-iris-architecture.png index 952908ba7158d9f2a2f78a5e195fa4e5194e8ade..b1d73a72f3604ab030aa5fc56c209ee897e6d0a8 100644 GIT binary patch literal 43325 zcmeFZWmuJ8(>6|brveg+w1i42A*BM+4Vw}H>5`C6DFH>16hsAtO^U*%VN=p5-E5F< z>He>c;4hw+`~TdZ-Y>6*U-UTkb**dGtaE10IcJ3`E8Zi-qrpQ#K_QgAFRhA#f^Lt3 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| 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/tests/test_iris.py b/tests/test_iris.py index 5193abe..bda1d6a 100644 --- a/tests/test_iris.py +++ b/tests/test_iris.py @@ -176,7 +176,7 @@ def nln_binary_iris_1(type, x, training: bool): Source: https://arxiv.org/pdf/1804.01508.pdf """ -# mean: 93.88, sem: 0.12, min: 80.00, max: 100.00, 5%: 86.67, 95%: 100.00 +# mean: 93.89, sem: 0.12, min: 80.00, max: 100.00, 5%: 86.67, 95%: 100.00 def nln_binary_iris(type, x, training: bool): dtype = jax.numpy.float64 y = hard_vmap.vmap(type)((lambda x: 1 - x, lambda x: 1 - x, lambda x: symbolic_primitives.symbolic_not(x)))(x) @@ -185,7 +185,7 @@ def nln_binary_iris(type, x, training: bool): x = hard_and.and_layer(type)( layer_size, dtype=dtype, - weights_init=initialization.initialize_bernoulli(0.0, 0.3, 0.501), + weights_init=initialization.initialize_uniform_range(0.4, 0.4), )(x) x = hard_dropout.hard_dropout(type)( rate=0.05, From fd11ccf49e3186c4c1cdd96c834b69ecf5ce1e79 Mon Sep 17 00:00:00 2001 From: Ian Wright Date: Fri, 24 Mar 2023 16:42:20 +0000 Subject: [PATCH 056/113] more --- docs/db.bib | 265 +++++++++++++++++++++++ docs/db.tex | 50 ++++- docs/noisy-xor-architecture.png | Bin 32025 -> 31980 bytes docs/papers/1705.11040.pdf | Bin 563892 -> 0 bytes docs/papers/1905.11885.pdf | Bin 1034619 -> 0 bytes docs/papers/2002.06100.pdf | Bin 4072464 -> 0 bytes docs/papers/PAYANI-DISSERTATION-2020.pdf | Bin 2408395 -> 0 bytes docs/papers/minimax-algebra.pdf | Bin 690025 -> 0 bytes 8 files changed, 304 insertions(+), 11 deletions(-) create mode 100644 docs/db.bib delete mode 100644 docs/papers/1705.11040.pdf delete mode 100644 docs/papers/1905.11885.pdf delete mode 100644 docs/papers/2002.06100.pdf delete mode 100644 docs/papers/PAYANI-DISSERTATION-2020.pdf delete mode 100644 docs/papers/minimax-algebra.pdf diff --git a/docs/db.bib b/docs/db.bib new file mode 100644 index 0000000..e17962d --- /dev/null +++ b/docs/db.bib @@ -0,0 +1,265 @@ +@misc{granmo18, + doi = {10.48550/ARXIV.1804.01508}, + + url = {https://arxiv.org/abs/1804.01508}, + + author = {Granmo, Ole-Christoffer}, + + keywords = {Artificial Intelligence (cs.AI), Computer Vision and Pattern Recognition (cs.CV), Machine Learning (cs.LG), FOS: Computer and information sciences, FOS: Computer and information sciences}, + + title = {The {T}setlin Machine -- A Game Theoretic Bandit Driven Approach to Optimal Pattern Recognition with Propositional Logic}, + + publisher = {arXiv}, + + year = {2018}, + + copyright = {arXiv.org perpetual, non-exclusive license} +} + +@misc{noisy-xor-dataset, + author = {Granmo, Ole-Christoffer}, + title = {The noisy {XOR} dataset}, + howpublished = {GitHub repository}, + url = {https://github.com/cair/TsetlinMachine} +} + +@misc{binary-iris-dataset, + author = {Granmo, Ole-Christoffer}, + title = {The binary Iris dataset}, + howpublished = {GitHub repository}, + url = {https://github.com/cair/TsetlinMachine} +} + +@inproceedings{ + Liu2020On, + title={On the Variance of the Adaptive Learning Rate and Beyond}, + author={Liyuan Liu and Haoming Jiang and Pengcheng He and Weizhu Chen and Xiaodong Liu and Jianfeng Gao and Jiawei Han}, + booktitle={International Conference on Learning Representations}, + year={2020}, + url={https://openreview.net/forum?id=rkgz2aEKDr} +} + +@article{DBLP:journals/corr/BengioLC13, + author = {Yoshua Bengio and + Nicholas L{\'{e}}onard and + Aaron C. Courville}, + title = {Estimating or Propagating Gradients Through Stochastic Neurons for + Conditional Computation}, + journal = {CoRR}, + volume = {abs/1308.3432}, + year = {2013}, + url = {http://arxiv.org/abs/1308.3432}, + eprinttype = {arXiv}, + eprint = {1308.3432}, + timestamp = {Mon, 13 Aug 2018 16:47:35 +0200}, + biburl = {https://dblp.org/rec/journals/corr/BengioLC13.bib}, + bibsource = {dblp computer science bibliography, https://dblp.org} +} + +@article{rumelhart1986learning, + title={Learning representations by back-propagating errors}, + author={Rumelhart, David E and Hinton, Geoffrey E and Williams, Ronald J}, + journal={Nature}, + volume={323}, + number={6088}, + pages={533--536}, + year={1986}, + publisher={Nature Publishing Group} +} + +@inproceedings{10.5555/3104322.3104425, + author = {Nair, Vinod and Hinton, Geoffrey E.}, + title = {Rectified Linear Units Improve Restricted Boltzmann Machines}, + year = {2010}, + isbn = {9781605589077}, + publisher = {Omnipress}, + address = {Madison, WI, USA}, + abstract = {Restricted Boltzmann machines were developed using binary stochastic hidden units. These can be generalized by replacing each binary unit by an infinite number of copies that all have the same weights but have progressively more negative biases. The learning and inference rules for these "Stepped Sigmoid Units" are unchanged. They can be approximated efficiently by noisy, rectified linear units. Compared with binary units, these units learn features that are better for object recognition on the NORB dataset and face verification on the Labeled Faces in the Wild dataset. Unlike binary units, rectified linear units preserve information about relative intensities as information travels through multiple layers of feature detectors.}, + booktitle = {Proceedings of the 27th International Conference on International Conference on Machine Learning}, + pages = {807–814}, + numpages = {8}, + location = {Haifa, Israel}, + series = {ICML'10} +} + +@inproceedings{10.5555/3157382.3157557, + author = {Hubara, Itay and Courbariaux, Matthieu and Soudry, Daniel and El-Yaniv, Ran and Bengio, Yoshua}, + title = {Binarized Neural Networks}, + year = {2016}, + isbn = {9781510838819}, + publisher = {Curran Associates Inc.}, + address = {Red Hook, NY, USA}, + abstract = {We introduce a method to train Binarized Neural Networks (BNNs) - neural networks with binary weights and activations at run-time. At train-time the binary weights and activations are used for computing the parameter gradients. During the forward pass, BNNs drastically reduce memory size and accesses, and replace most arithmetic operations with bit-wise operations, which is expected to substantially improve power-efficiency. To validate the effectiveness of BNNs, we conducted two sets of experiments on the Torch7 and Theano frameworks. On both, BNNs achieved nearly state-of-the-art results over the MNIST, CIFAR-10 and SVHN datasets. We also report our preliminary results on the challenging ImageNet dataset. Last but not least, we wrote a binary matrix multiplication GPU kernel with which it is possible to run our MNIST BNN 7 times faster than with an unoptimized GPU kernel, without suffering any loss in classification accuracy. The code for training and running our BNNs is available on-line.}, + booktitle = {Proceedings of the 30th International Conference on Neural Information Processing Systems}, + pages = {4114–4122}, + numpages = {9}, + location = {Barcelona, Spain}, + series = {NIPS'16} +} + +@article{JMLR:v15:srivastava14a, + author = {Nitish Srivastava and Geoffrey Hinton and Alex Krizhevsky and Ilya Sutskever and Ruslan Salakhutdinov}, + title = {Dropout: A Simple Way to Prevent Neural Networks from Overfitting}, + journal = {Journal of Machine Learning Research}, + year = {2014}, + volume = {15}, + number = {56}, + pages = {1929--1958}, + url = {http://jmlr.org/papers/v15/srivastava14a.html} +} + +@inproceedings{10.5555/2969442.2969588, + author = {Courbariaux, Matthieu and Bengio, Yoshua and David, Jean-Pierre}, + title = {BinaryConnect: Training Deep Neural Networks with Binary Weights during Propagations}, + year = {2015}, + publisher = {MIT Press}, + address = {Cambridge, MA, USA}, + abstract = {Deep Neural Networks (DNN) have achieved state-of-the-art results in a wide range of tasks, with the best results obtained with large training sets and large models. In the past, GPUs enabled these breakthroughs because of their greater computational speed. In the future, faster computation at both training and test time is likely to be crucial for further progress and for consumer applications on low-power devices. As a result, there is much interest in research and development of dedicated hardware for Deep Learning (DL). Binary weights, i.e., weights which are constrained to only two possible values (e.g. -1 or 1), would bring great benefits to specialized DL hardware by replacing many multiply-accumulate operations by simple accumulations, as multipliers are the most space and power-hungry components of the digital implementation of neural networks. We introduce BinaryConnect, a method which consists in training a DNN with binary weights during the forward and backward propagations, while retaining precision of the stored weights in which gradients are accumulated. Like other dropout schemes, we show that BinaryConnect acts as regularizer and we obtain near state-of-the-art results with BinaryConnect on the permutation-invariant MNIST, CIFAR-10 and SVHN.}, + booktitle = {Proceedings of the 28th International Conference on Neural Information Processing Systems - Volume 2}, + pages = {3123–3131}, + numpages = {9}, + location = {Montreal, Canada}, + series = {NIPS'15} +} + +@ARTICLE {9026948, + author = {E. Wang and J. J. Davis and P. K. Cheung and G. A. Constantinides}, + journal = {IEEE Transactions on Computers}, + title = {LUTNet: Learning FPGA Configurations for Highly Efficient Neural Network Inference}, + year = {2020}, + volume = {69}, + number = {12}, + issn = {1557-9956}, + pages = {1795-1808}, + abstract = {Research has shown that deep neural networks contain significant redundancy, and thus that high classification accuracy can be achieved even when weights and activations are quantized down to binary values. Network binarization on FPGAs greatly increases area efficiency by replacing resource-hungry multipliers with lightweight XNOR gates. However, an FPGA's fundamental building block, the K-LUT, is capable of implementing far more than an XNOR: it can perform any K-input Boolean operation. Inspired by this observation, we propose LUTNet, an end-to-end hardware-software framework for the construction of area-efficient FPGA-based neural network accelerators using the native LUTs as inference operators. We describe the realization of both unrolled and tiled LUTNet architectures, with the latter facilitating smaller, less power-hungry deployment over the former while sacrificing area and energy efficiency along with throughput. For both varieties, we demonstrate that the exploitation of LUT flexibility allows for far heavier pruning than possible in prior works, resulting in significant area savings while achieving comparable accuracy. Against the state-of-the-art binarized neural network implementation, we achieve up to twice the area efficiency for several standard network models when inferencing popular datasets. We also demonstrate that even greater energy efficiency improvements are obtainable.}, + keywords = {table lookup;deep learning;field programmable gate arrays;neural networks;logic gates;random access memory}, + doi = {10.1109/TC.2020.2978817}, + publisher = {IEEE Computer Society}, + address = {Los Alamitos, CA, USA}, + month = {dec} +} + +@InProceedings{10.1007/978-3-319-46493-0_32, + author="Rastegari, Mohammad + and Ordonez, Vicente + and Redmon, Joseph + and Farhadi, Ali", + editor="Leibe, Bastian + and Matas, Jiri + and Sebe, Nicu + and Welling, Max", + title="XNOR-Net: ImageNet Classification Using Binary Convolutional Neural Networks", + booktitle="Computer Vision -- ECCV 2016", + year="2016", + publisher="Springer International Publishing", + address="Cham", + pages="525--542", + abstract="We propose two efficient approximations to standard convolutional neural networks: Binary-Weight-Networks and XNOR-Networks. In Binary-Weight-Networks, the filters are approximated with binary values resulting in 32{\$}{\$}{\backslash}times {\$}{\$}{\texttimes}memory saving. In XNOR-Networks, both the filters and the input to convolutional layers are binary. XNOR-Networks approximate convolutions using primarily binary operations. This results in 58{\$}{\$}{\backslash}times {\$}{\$}{\texttimes}faster convolutional operations (in terms of number of the high precision operations) and 32{\$}{\$}{\backslash}times {\$}{\$}{\texttimes}memory savings. XNOR-Nets offer the possibility of running state-of-the-art networks on CPUs (rather than GPUs) in real-time. Our binary networks are simple, accurate, efficient, and work on challenging visual tasks. We evaluate our approach on the ImageNet classification task. The classification accuracy with a Binary-Weight-Network version of AlexNet is the same as the full-precision AlexNet. We compare our method with recent network binarization methods, BinaryConnect and BinaryNets, and outperform these methods by large margins on ImageNet, more than {\$}{\$}16{\backslash},{\backslash}{\%}{\$}{\$}16{\%}in top-1 accuracy. Our code is available at: http://allenai.org/plato/xnornet.", + isbn="978-3-319-46493-0" +} + +@article{QIN2020107281, + title = {Binary neural networks: A survey}, + journal = {Pattern Recognition}, + volume = {105}, + pages = {107281}, + year = {2020}, + issn = {0031-3203}, + doi = {https://doi.org/10.1016/j.patcog.2020.107281}, + url = {https://www.sciencedirect.com/science/article/pii/S0031320320300856}, + author = {Haotong Qin and Ruihao Gong and Xianglong Liu and Xiao Bai and Jingkuan Song and Nicu Sebe}, + keywords = {Binary neural network, Deep learning, Model compression, Network quantization, Model acceleration}, + abstract = {The binary neural network, largely saving the storage and computation, serves as a promising technique for deploying deep models on resource-limited devices. However, the binarization inevitably causes severe information loss, and even worse, its discontinuity brings difficulty to the optimization of the deep network. To address these issues, a variety of algorithms have been proposed, and achieved satisfying progress in recent years. In this paper, we present a comprehensive survey of these algorithms, mainly categorized into the native solutions directly conducting binarization, and the optimized ones using techniques like minimizing the quantization error, improving the network loss function, and reducing the gradient error. We also investigate other practical aspects of binary neural networks such as the hardware-friendly design and the training tricks. Then, we give the evaluation and discussions on different tasks, including image classification, object detection and semantic segmentation. Finally, the challenges that may be faced in future research are prospected.} +} + +@inproceedings{ + dong2018neural, + title={Neural Logic Machines}, + author={Honghua Dong and Jiayuan Mao and Tian Lin and Chong Wang and Lihong Li and Denny Zhou}, + booktitle={International Conference on Learning Representations}, + year={2019}, + url={https://openreview.net/forum?id=B1xY-hRctX}, +} + +@article{10.5555/3241691.3241692, + author = {Evans, Richard and Grefenstette, Edward}, + title = {Learning Explanatory Rules from Noisy Data}, + year = {2018}, + issue_date = {January 2018}, + publisher = {AI Access Foundation}, + address = {El Segundo, CA, USA}, + volume = {61}, + number = {1}, + issn = {1076-9757}, + abstract = {Artificial Neural Networks are powerful function approximators capable of modelling solutions to a wide variety of problems, both supervised and unsupervised. As their size and expressivity increases, so too does the variance of the model, yielding a nearly ubiquitous over_tting problem. Although mitigated by a variety of model regularisation methods, the common cure is to seek large amounts of training data--which is not necessarily easily obtained--that sufficiently approximates the data distribution of the domain we wish to test on. In contrast, logic programming methods such as Inductive Logic Programming offer an extremely data-efficient process by which models can be trained to reason on symbolic domains. However, these methods are unable to deal with the variety of domains neural networks can be applied to: they are not robust to noise in or mislabelling of inputs, and perhaps more importantly, cannot be applied to non-symbolic domains where the data is ambiguous, such as operating on raw pixels. In this paper, we propose a Differentiable Inductive Logic framework, which can not only solve tasks which traditional ILP systems are suited for, but shows a robustness to noise and error in the training data which ILP cannot cope with. Furthermore, as it is trained by backpropagation against a likelihood objective, it can be hybridised by connecting it with neural networks over ambiguous data in order to be applied to domains which ILP cannot address, while providing data efficiency and generalisation beyond what neural networks on their own can achieve.}, + journal = {J. Artif. Int. Res.}, + month = {jan}, + pages = {1–64}, + numpages = {64} +} + +@Book{BreiFrieStonOlsh84, + Title = {Classification and Regression Trees}, + Author = {Leo Breiman, Jerome Friedman, Charles J. Stone, R.A. Olshen}, + Publisher = {Chapman and Hall/CRC}, + Year = {1984} +} + +@INPROCEEDINGS{598994, + author={Tin Kam Ho}, + booktitle={Proceedings of 3rd International Conference on Document Analysis and Recognition}, + title={Random decision forests}, + year={1995}, + volume={1}, + number={}, + pages={278-282 vol.1}, + doi={10.1109/ICDAR.1995.598994}} + +@book{koza1992genetic, + title={Genetic Programming: On the Programming of Computers by Means of Natural Selection}, + author={Koza, J.R.}, + isbn={9780262111706}, + lccn={92025785}, + series={A Bradford book}, + url={https://books.google.co.uk/books?id=Bhtxo60BV0EC}, + year={1992}, + publisher={Bradford} +} + +@inproceedings{NEURIPS2019_d8c24ca8, + author = {Cuturi, Marco and Teboul, Olivier and Vert, Jean-Philippe}, + booktitle = {Advances in Neural Information Processing Systems}, + editor = {H. Wallach and H. Larochelle and A. Beygelzimer and F. d\textquotesingle Alch\'{e}-Buc and E. Fox and R. Garnett}, + pages = {}, + publisher = {Curran Associates, Inc.}, + title = {Differentiable Ranking and Sorting using Optimal Transport}, + url = {https://proceedings.neurips.cc/paper_files/paper/2019/file/d8c24ca8f23c562a5600876ca2a550ce-Paper.pdf}, + volume = {32}, + year = {2019} +} + +@article{VANKRIEKEN2022103602, + title = {Analyzing Differentiable Fuzzy Logic Operators}, + journal = {Artificial Intelligence}, + volume = {302}, + pages = {103602}, + year = {2022}, + issn = {0004-3702}, + doi = {https://doi.org/10.1016/j.artint.2021.103602}, + url = {https://www.sciencedirect.com/science/article/pii/S0004370221001533}, + author = {Emile {van Krieken} and Erman Acar and Frank {van Harmelen}}, + keywords = {Fuzzy logic, Neural-symbolic AI, Learning with constraints}, + abstract = {The AI community is increasingly putting its attention towards combining symbolic and neural approaches, as it is often argued that the strengths and weaknesses of these approaches are complementary. One recent trend in the literature is weakly supervised learning techniques that employ operators from fuzzy logics. In particular, these use prior background knowledge described in such logics to help the training of a neural network from unlabeled and noisy data. By interpreting logical symbols using neural networks, this background knowledge can be added to regular loss functions, hence making reasoning a part of learning. We study, both formally and empirically, how a large collection of logical operators from the fuzzy logic literature behave in a differentiable learning setting. We find that many of these operators, including some of the most well-known, are highly unsuitable in this setting. A further finding concerns the treatment of implication in these fuzzy logics, and shows a strong imbalance between gradients driven by the antecedent and the consequent of the implication. Furthermore, we introduce a new family of fuzzy implications (called sigmoidal implications) to tackle this phenomenon. Finally, we empirically show that it is possible to use Differentiable Fuzzy Logics for semi-supervised learning, and compare how different operators behave in practice. We find that, to achieve the largest performance improvement over a supervised baseline, we have to resort to non-standard combinations of logical operators which perform well in learning, but no longer satisfy the usual logical laws.} +} + +@phdthesis{DBLP:phd/basesearch/Payani20, + author = {Ali Payani}, + title = {Differentiable neural logic networks and their application onto inductive + logic programming}, + school = {Georgia Institute of Technology, Atlanta, GA, {USA}}, + year = {2020}, + url = {https://hdl.handle.net/1853/62833}, + timestamp = {Wed, 04 May 2022 13:00:04 +0200}, + biburl = {https://dblp.org/rec/phd/basesearch/Payani20.bib}, + bibsource = {dblp computer science bibliography, https://dblp.org} +} \ No newline at end of file diff --git a/docs/db.tex b/docs/db.tex index a334d12..074c822 100644 --- a/docs/db.tex +++ b/docs/db.tex @@ -57,7 +57,7 @@ \section{Introduction} This paper proposes a new approach to mitigate these drawbacks. The main idea is to define a type of neural network, called a $\partial \mathbb{B}$ net, which has two aspects: a soft net, which is a differentiable real-valued function, and a hard net, which is a non-differentiable, discrete function. Both aspects are semantically equivalent. We train the soft net as normal, using backpropagation, then `harden' the learned weights to boolean values and bind them with the hard net to yield a discrete function with identical predictive performance (see figure \ref{fig:main-idea}). In consequence, interpreting and verifying a $\partial \mathbb{B}$ net is relatively less difficult. The bias towards learning discrete functions reduces variance in some domains. And boolean-valued, 1-bit weights increase the memory-efficiency of trained models. -The main contributions of this work are (i) defining novel activation functions that `harden' to semantically equivalent discrete functions, (ii) defining novel network architectures to effectively learn discrete functions that solve multi-class classification problems, and (iii) experiments that demonstrate $\partial \mathbb{B}$ nets compete with existing approaches in terms of predictive performance yet yield considerably smaller models. +The main contributions of this work are (i) defining novel activation functions that `harden' to semantically equivalent discrete functions without loss of predictive performance, (ii) defining novel network architectures to effectively learn discrete functions that solve multi-class classification problems, and (iii) experiments that demonstrate $\partial \mathbb{B}$ nets compete with existing approaches in terms of predictive performance yet yield considerably smaller models. This paper examines related work (section \ref{sec:related-work}), (ii) etc. @@ -68,10 +68,30 @@ \section{Introduction} \label{fig:main-idea} \end{figure} - \section{Related work}\label{sec:related-work} -todo: can Tsetlin machines be chained in differentiable architectures? +Methods to learn complex, discrete boolean functions can be broadly categorized as either non-differentiable or differentiable. + +Non-differentiable approaches include boolean-valued decision trees \citep{BreiFrieStonOlsh84}, random forests \citep{598994} and genetic programming \citep{koza1992genetic}. Tsetlin machines \cite{granmo18} represent propositional formulae by collections of Tsetlin Automata with integer weights that learn by positive and negative feedback. Tsetlin Machines are not end-to-end differentiable because feedback mechanism uses a hard threshold function. + +These models can be highly interpretable but can fail to capture complex relationships between inputs and outputs. + +Differentiable approaches include differentiable Inductive Logic Programming \citep{10.5555/3241691.3241692} learns first-order logic rules using gradient descent. Neural Logic Machines \citep{dong2018neural} also learn first-order logic rules. First-order logic is more expressive than propositional logic. However, don't scale to large problem instances. + +These approaches also aim to learn discrete functions while maintaining end-to-end differentiability. + +Binarization reduces model size and computation cost while attempting to maintain accuracy by reducing real-valued weights and activations to binary values. The binarized neurons typically implement boolean majority and therefore the entire network is a boolean function. For example, BinaryConnect \citep{10.5555/2969442.2969588} optimizes a continuous relaxation of binary weights during training. However, training is relatively difficult because binary weights are intrinsically not differentiable. Binarization tends to reduce accuracy due to the loss of information \citep{QIN2020107281}. So research has focussed on avoiding accuracy loss (e.g. XNOR-Net \citep{10.1007/978-3-319-46493-0_32} employs a real-valued scaling factor to improve accuracy). + +LUTNet \citep{9026948} + +We don't aim to learn fuzzy logic, but hard logic. See \cite{VANKRIEKEN2022103602} for comparable approaches to activation functions that represent logical operations (Godel, product etc.) + +\cite{DBLP:phd/basesearch/Payani20} for an example of similar but not hard-equivalent activations. + +In summary, while there has been significant research in the area of binary-valued neural networks and boolean functions, there is still much to be explored in terms of developing models that can learn complex boolean functions while maintaining high predictive accuracy. + +In this paper, we propose a new kind of neural network that is end-to-end differentiable, yet learns an arbitrarily complex, discrete boolean function without losing predictive accuracy. We require a separate full-precision copy of the weights for efficient training. Avoids need for feature engineering required by direct boolean function learning methods. + \section{$\partial\mathbb{B}$ nets} @@ -270,6 +290,8 @@ \subsection{Differentiable majority} \end{equation*} which is hard-equivalent to $\operatorname{Maj}({\bf x})$ (see proposition \ref{prop:majority}). Note that $\partial\!\operatorname{Maj}$ is differentiable almost everywhere and gradient-rich (see figure \ref{fig:majority-plot}). If $\operatorname{sort}$ is quicksort then the the average time-complexity of $\partial\!\operatorname{Maj}$ is $\mathcal{O}(n\log{}n)$, which makes $\partial\!\operatorname{Maj}$ more expensive than $\partial_{\neg}$, $\partial_{\wedge}$, $\partial_{\vee}$ and $\partial_{\Rightarrow}$ at training time. However, in the hard $\partial\mathbb{B}$ net we efficiently implement $\operatorname{Maj}$ as a discrete program that simply checks if the majority of bits are high. +TODO: compare to \cite{NEURIPS2019_d8c24ca8} + \subsection{Differentiable counting} A boolean counting function $f({\bf x})$ has the value $1$ if a counting predicate, $c({\bf x})$, holds over its $n$ inputs. For example, $\partial\!\operatorname{Maj}({\bf x})$ is a boolean counting function where $c({\bf x}) := |\{x_{i} : x_{i} = 1 \}| \geq \lceil \frac{n}{2} \rceil$. We aim to construct a differentiable analogue of $\operatorname{count}({\bf x}, k)$ where $c({\bf x}) := |\{x_{i} : x_{i} = 1 \}| = k$ (i.e. `exactly $k$ high'). This is useful for multiclass classification problems where we interpret $k$ as a class prediction. @@ -321,14 +343,14 @@ \subsection{Boolean logic layers} \end{equation*} where ${\bf x}$ is a soft-bit input vector, ${\bf W}$ is a weight matrix and $n$ is the layer width. -A $\partial_{\wedge}\!\operatorname{Neuron}$ learns to logically $\wedge$ a subset of the elements of its input vector: +A $\partial_{\wedge}\!\operatorname{Neuron}$ learns to logically $\wedge$ a subset of its input vector: \begin{equation*} \begin{aligned} \partial_{\wedge}\!\operatorname{Neuron}: [0,1]^{n} \times [0,1]^{n} &\to [0,1], \\ ({\bf w}, {\bf x}) &\mapsto \min(\partial_{\Rightarrow}\!(w_{1}, x_{1}), \dots, \partial_{\Rightarrow}\!(w_{n}, x_{n}))\text{,} \end{aligned} \end{equation*} -where ${\bf w}$ is a weight vector. A $\partial_{\wedge}\!\operatorname{Neuron}$ maps $n$ inputs to a single soft-bit output. A $\partial_{\wedge}\!\operatorname{Layer}$ of $n$ neurons then maps $m$ inputs to $n$ outputs, where each output is the logical $\wedge$ of some subset of the input vector. In other words, a $\partial_{\wedge}\!\operatorname{Layer}$ of width $n$ can learn up to $n$ different conjunctions of subsets of its input. +where ${\bf w}$ is a weight vector. Each $\partial_{\Rightarrow}(w_{i},x_{i})$ learns to include or exclude $x_{i}$ from the conjunction depending on weight $w_{i}$. For example, if $w_{i}>0.5$ then $x_{i}$ affects the value of the conjunction because $\partial_{\Rightarrow}(w_{i},x_{i})$ passes-through a soft-bit that is high if $x_{i}$ is high, and low otherwise; but if $w_{i} \leq 0.5$ then $x_{i}$ does not affect the conjunction because $\partial_{\Rightarrow}(w_{i},x_{i})$ always passes-through a high soft-bit. A $\partial_{\wedge}\!\operatorname{Layer}$ of width $n$ learns up to $n$ different conjunctions of subsets of its input (of whatever size). A $\partial_{\vee}\!\operatorname{Neuron}$ is defined similarly: \begin{equation*} @@ -337,7 +359,7 @@ \subsection{Boolean logic layers} ({\bf w}, {\bf x}) &\mapsto \max(\partial_{\wedge}\!(w_{1}, x_{1}), \dots, \partial_{\wedge}\!(w_{n}, x_{n}))\text{.} \end{aligned} \end{equation*} -In consequence, a $\partial_{\vee}\!\operatorname{Layer}$ of width $n$ can learn up to $n$ different disjunctions of subsets of its input. +Each $\partial_{\wedge}(w_{i},x_{i})$ learns to include or exclude $x_{i}$ from the disjunction depending on weight $w_{i}$. For example, if $w_{i}>0.5$ then $x_{i}$ affects the value of the conjunction because $\partial_{\wedge}(w_{i},x_{i})$ passes-through a soft-bit that is high if $x_{i}$ is high, and low otherwise; but if $w_{i} \leq 0.5$ then $x_{i}$ does not affect the conjunction because $\partial_{\Rightarrow}(w_{i},x_{i})$ always passes-through a low soft-bit. A $\partial_{\vee}\!\operatorname{Layer}$ of width $n$ learns up to $n$ different disjunctions of subsets of its input (of whatever size). We compose $\partial_{\neg}$, $\partial_{\wedge}$ and $\partial_{\vee}$ layers to learn boolean formulae of arbitrary width and depth. @@ -370,13 +392,17 @@ \subsection{Classification layers} \end{equation*} At train time $\partial\!\operatorname{dropout}$ randomly negates soft-bit values with probability $p$. At test time, and in the hard net, $\partial\!\operatorname{dropout}$ is a $\operatorname{nop}$. +\section{Implementation} + +The $\partial\mathbb{B}$ net library is open-source and available at {\small \url{https://github.com/Z80coder/db-nets}}. + \section{Experiments} \subsection{Hardening} \subsection{Binary Iris} -\begin{figure}[t] +\begin{figure}[t!] \centering \includegraphics[width=1.0\textwidth]{binary-iris-architecture.png} \caption{{\em A $\partial\mathbb{B}$ net for the binary Iris problem}. The net concatenates the soft-bit input, ${\bf x}$ (length 16), with its negation, ${\bf 1 - x}$, and supplies the resulting vector (length 32) to a $\partial_{\wedge}\!\operatorname{Layer}$ (width 59), a $\partial\!\operatorname{dropout}$ layer for improved generalisation, a $\partial\!\operatorname{count-hot}$ layer that generates a 1-hot vector (width 60) that is reduced by $\operatorname{max}$ to a 1-hot vector of 3 classification bits. A final $\partial\!\operatorname{harden}$ ensures the loss function is a function of hard bits. The net's weights, once hardened, consume $236$ bytes.} @@ -385,7 +411,7 @@ \subsection{Binary Iris} The Iris dataset has 150 examples with 4 inputs (sepal length and width, and petal length and width), and 3 labels ({\em setosa}, {\em versicolour}, and {\em virginica}). We use the binary version of the Iris dataset \citep{binary-iris-dataset} where each input float is represented by 4 bits. We perform 1000 experiments, each with a different random seed. Each experiment randomly partitions the data into 20\% training and 80\% test sets. -We initialize the network described in figure \ref{fig:binary-iris-architecture} using the XX policy and train for 1000 epochs with the RAdam optimizer \citep{Liu2020On} and softmax cross-entropy loss. We measure the accuracy of the final net to avoid hand-picking the best configuration. Table \ref{tab:binary-iris-results} compares the $\delta\mathbb{B}$ net against other classifiers \citep{granmo18}. Naive Bayes performs the worst. The Tsetlin machine performs best on this problem, with the $\partial\mathbb{B}$ net second, notably beating a standard multilayer neural network. +We initialize the network, described in figure \ref{fig:binary-iris-architecture}, with all weights $w_{i} = 0.3$ and train for 1000 epochs with the RAdam optimizer \citep{Liu2020On} and softmax cross-entropy loss. We measure the accuracy of the final net to avoid hand-picking the best configuration. Table \ref{tab:binary-iris-results} compares the $\delta\mathbb{B}$ net against other classifiers \citep{granmo18}. Naive Bayes performs the worst. The Tsetlin machine performs best on this problem, with the $\partial\mathbb{B}$ net second. \begin{table}[t] \centering @@ -399,14 +425,14 @@ \subsection{Binary Iris} \multicolumn{1}{|l|}{SVM} & \multicolumn{1}{l|}{93.6 +/- 0.3} & \multicolumn{1}{l|}{86.7} & \multicolumn{1}{l|}{100.0} & \multicolumn{1}{l|}{76.7} & \multicolumn{1}{l|}{100.0} \\ \hline \multicolumn{1}{|l|}{naive Bayes} & \multicolumn{1}{l|}{91.6 +/- 0.3} & \multicolumn{1}{l|}{83.3} & \multicolumn{1}{l|}{96.7} & \multicolumn{1}{l|}{70.0} & \multicolumn{1}{l|}{100.0} \\ \hline \end{tabular} - \caption{{\em Ranked binary iris results}} + \caption{{\em Ranked binary Iris results}} \label{tab:binary-iris-results} \end{table} \subsection{Noisy XOR} -\begin{figure}[t] +\begin{figure}[t!] \centering \includegraphics[width=1.0\textwidth]{noisy-xor-architecture.png} \caption{{\em A $\partial\mathbb{B}$ net for the noisy xor problem}. The net concatenates the soft-bit input, ${\bf x}$ (length 12), with its negation, ${\bf 1 - x}$, and supplies the resulting vector (length 24) to a $\partial_{\wedge}\!\!\operatorname{Layer}$ (width 32), $\partial_{\vee}\!\!\operatorname{Layer}$ (width 32), $\partial_{\neg} \!\operatorname{Layer}$ (width 16), and a final $\partial\!\operatorname{Maj}$ to produce a single soft-bit $y \in [0,1]$ (to predict odd parity) and its negation $1-y$ (to predict even parity). The net's weights, once hardened, consume $288$ bytes.} @@ -415,7 +441,9 @@ \subsection{Noisy XOR} The noisy XOR dataset \citep{noisy-xor-dataset} is an adversarial parity problem with noisy non-informative features. The dataset consists of 10K examples with 12 boolean inputs and a target label (where 0 = odd and 1 = even) that is a XOR function of 2 inputs. The remaining 10 inputs are entirely random. We train on 50\% of the data where, additionally, 40\% of the labels are inverted. -We randomly initialize the network described in figure \ref{fig:noisy-xor-architecture} using the XX policy and then trained for 2000 epochs with the RAdam optimizer and softmax cross-entropy loss. We measure the accuracy of the final net on the test data to avoid hand-picking the best configuration. Table \ref{tab:noisy-xor-results} compares the $\partial\mathbb{B}$ net against other classifiers \citep{granmo18}. The high noise causes logistic regression and naive Bayes to randomly guess. The SVM hardly performs better. In constrast, the multilayer neural network, Tsetlin machine \citep{granmo18}, and $\partial\mathbb{B}$ net all successfully learn the underlying XOR signal. The Tsetlin machine performs best on this problem, with the $\partial\mathbb{B}$ net second. +We initialize the network described in figure \ref{fig:noisy-xor-architecture} with random weights distributed close to the hard threshold at $1/2$ (i.e. in the $\partial_{\wedge}\!\operatorname{Layer}$, $w_{i} = 0.501 \times b + 0.3 \times (1-b)$ where $b \sim \operatorname{Bernoulli}(0.01)$; in the $\partial_{\vee}\!\operatorname{Layer}$, $w_{i} = 0.7 \times b + 0.499 \times (1-b)$ where $b \sim \operatorname{Bernoulli}(0.99)$); and in the $\partial_{\neg}\!\operatorname{Layer}$, $w_{i} \sim \operatorname{Uniform}(0.499, 0.501)$. We train for 2000 epochs with the RAdam optimizer and softmax cross-entropy loss. + +We measure the accuracy of the final net on the test data to avoid hand-picking the best configuration. Table \ref{tab:noisy-xor-results} compares the $\partial\mathbb{B}$ net against other classifiers \citep{granmo18}. The high noise causes logistic regression and naive Bayes to randomly guess. The SVM hardly performs better. In constrast, the multilayer neural network, Tsetlin machine, and $\partial\mathbb{B}$ net all successfully learn the underlying XOR signal. 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Sep 17 00:00:00 2001 From: Ian Wright Date: Tue, 21 Mar 2023 12:40:03 +0000 Subject: [PATCH 047/113] more --- docs/db.tex | 149 ++++++------ docs/margin-trick.png | Bin 29935 -> 29607 bytes docs/proofs.nb | 545 ++++++++++++++++++++---------------------- 3 files changed, 345 insertions(+), 349 deletions(-) diff --git a/docs/db.tex b/docs/db.tex index f74cdcf..5e88b76 100644 --- a/docs/db.tex +++ b/docs/db.tex @@ -61,7 +61,7 @@ \section{Introduction} \begin{figure}[h] \centering - \includegraphics[width=0.9\textwidth]{db-net.png} + \includegraphics[width=1.0\textwidth]{db-net.png} \caption{{\em Learning discrete functions with a $\partial\mathbb{B}$ net.} A $\partial \mathbb{B}$ net specifies (i) a differentiable neural network that is hard-equivalent to (ii) a non-differentiable discrete function. The neural network is trained as normal with backpropagation to yield a set of real weights. The real weights are hardened to boolean values and then bound with the discrete function. The result is a learned discrete function that performs identically to the trained network.} \label{fig:main-idea} \end{figure} @@ -99,105 +99,103 @@ \section{$\partial\mathbb{B}$ nets} \begin{equation*} \operatorname{harden}(f(\operatorname{harden}({\bf x}))) = g(\operatorname{harden}({\bf x})) \end{equation*} -for all ${\bf x} \in [0,1]^{n}$. +for all ${\bf x} \in [0,1]^{n}$. For shorthand write $f \equiv g$. \end{definition} -$\partial \mathbb{B}$ nets are arbitrary compositions of differentiable functions that are hard-equivalent to boolean functions (and natural generalisations). - -Neural networks are typically composed of nonlinear activation functions (for representational generality) that are strictly monotonic (so gradients always exist that link changes in inputs to outputs) and differentiable (so gradients reliably represent the local loss surface). Activation functions that are only monotonic (so some gradients are zero) and differentiable almost everywhere (so some gradients are undefined) can also work, e.g. RELU \citep{10.5555/3104322.3104425}. $\partial \mathbb{B}$ nets are also composed of `activation' functions that satisfy these properties. But $\partial \mathbb{B}$ net activation functions must also satisfy the additional requirement of hard-equivalence. +Neural networks are typically composed of nonlinear activation functions (for representational generality) that are strictly monotonic (so gradients always exist that link changes in inputs to outputs) and differentiable (so gradients reliably represent the local loss surface). Activation functions that are monotonic but not strictly so (and therefore some gradients are zero) and differentiable almost everywhere (and therefore some gradients are undefined) also work, e.g. RELU \citep{10.5555/3104322.3104425}. $\partial \mathbb{B}$ nets are arbitrary compositions of `activation' functions that also satisfy these properties but in addition are hard-equivalent to boolean functions (and natural generalisations). \begin{figure}[t] \centering - \includegraphics[trim=0pt 0pt 0pt 0pt, clip, width=0.975\textwidth]{logic-gates.png} + \includegraphics[trim=0pt 0pt 0pt 0pt, clip, width=1.0\textwidth]{logic-gates.png} \caption{{\em Gradient-rich versus gradient-sparse differentiable boolean functions.} Each column contains contour plots of functions $f(x,y)$ that are hard-equivalent to a boolean function (one of $\neg(x \oplus y)$, $x \wedge y$, $x \vee y$, or $x \Rightarrow y$). Every function is continuous and differentiable almost everywhere (white lines indicate non-continuous derivatives). The upper plots are gradient-sparse, where vertical and horizontal contours indicate the function is constant with respect to one of its inputs, i.e. $\partial f/\partial y = 0$ or $\partial f/\partial x = 0$. The lower plots are gradient-rich, where the curved contours indicate the function always varies with respect to any of its inputs, i.e. $\partial f/\partial y \neq 0$ and $\partial f/\partial x \neq 0$. $\partial \mathbb{B}$ nets use gradient-rich functions to ensure that error is always backpropagated to all inputs.} \label{fig:gradient-rich} \end{figure} \subsection{Learning to negate} -We want to learn whether to negate a boolean value, $x$, or simply leave it unaltered. We represent this decision by a boolean weight, $w$, where low $w$ means negate and high $w$ means do nothing. The boolean function that meets this requirement is $\neg(x \oplus w)$. However, this function is not differentiable. We therefore define the differentiable function, +We aim to learn to negate a boolean value, $x$, or simply leave it unaltered. Represent this decision by a boolean weight, $w$, where low $w$ means negate and high $w$ means do nothing. The boolean function that meets this requirement is $\neg(x \oplus w)$. However, this function is not differentiable. We therefore define the differentiable function, \begin{equation*} \begin{aligned} - \partial\text{NOT}: [0, 1]^{2} &\to [0,1], \\ + \partial \neg: [0, 1]^{2} &\to [0,1], \\ (w, x) &\mapsto 1 - w + x (2w - 1)\text{,} \end{aligned} \end{equation*} -which is hard-equivalent to the boolean function $\neg(x \oplus w)$ (see proposition \ref{prop:not}). +where $\partial \neg(w, x) \equiv \neg(x \oplus w)$ (see proposition \ref{prop:not}). -Note that $\operatorname{min}$ is hard-equivalent to $\wedge$ and $\operatorname{max}$ is hard-equivalent to $\vee$. So the function $\operatorname{max}(\operatorname{min}(w, x), \operatorname{min}(1-w, 1-x))$ is also hard-equivalent to $\neg(x \oplus w)$. But $\partial\text{NOT}$, in contrast, is a gradient-rich function that always backpropagates error to all its inputs (see figure \ref{fig:gradient-rich}). +Product logics, where for example $f(x,y) = x y$ is as a soft version of $x \wedge y$, although hard-equivalent at extreme values, e.g. $f(1,1)=1$ and $f(0,1)=0$, are not hard-equivalent at intermediate values, e.g. $f(0.6, 0.6) = 0.36$. G\"{o}del-style $\operatorname{min}$ and $\operatorname{max}$ functions, although hard-equivalent over the entire soft-bit range, i.e. $\operatorname{min}(x,y) \equiv x \wedge y$ and $\operatorname{min}(x,y) \equiv x \vee y$, are gradient-sparse in the sense that their outputs are not always a function of all their inputs, e.g. $\frac{\partial}{\partial x} \operatorname{max}(x,y) = 0$ when $(x,y)=(0.1, 0.9)$. So although the composite function $\operatorname{max}(\operatorname{min}(w, x), \operatorname{min}(1-w, 1-x))$ is differentiable and $\equiv \neg(x \oplus w)$ it does not always backpropagate error to its inputs. In contrast, $\partial \neg$ is a gradient-rich function that always backpropagates error to its inputs (see figure \ref{fig:gradient-rich}). -\subsection{Differentiable $\wedge$} +\subsection{Margin packing} -As stated $\operatorname{min}(x, y)$ is differentiable and hard-equivalent to $x \wedge y$. However, it is gradient-sparse, e.g. $\frac{\partial}{\partial x} \operatorname{min}(x,y) = 0$ when $(x,y)=(0.9,0.1)$. Other soft logic functions, such as $f(x,y) = x y$, are not hard-equivalent at intermediate soft-bit values, e.g. $f(0.6, 0.6) = 0.36$. To construct a gradient-rich, hard-equivalent function we first observe that $\operatorname{min}(x,y)$ essentially selects one of $x$ or $y$ as a `representative soft-bit' that is guaranteed to be hard-equivalent to $x \wedge y$. The representative bit, whether low or high, has a corresponding margin up to the threshold value $1/2$. +Say we aim to construct a differentiable analogue of $x \wedge y$. Note that $\operatorname{min}(x,y)$ essentially selects one of $x$ or $y$ as a `representative soft-bit' that is guaranteed hard-equivalent to $x \wedge y$. However, by selecting only one of $x$ or $y$ then $\operatorname{min}$ is also guaranteed to be gradient-sparse. We define a `margin packing' method to avoid this dilemma. + +The main idea of margin packing is (i) select a representative bit that is hard-equivalent to the target discrete function, and then (ii) pack a fraction of the margin between the representative bit and the hard threshold $1/2$ with gradient-rich information. The result is an augmented bit that is a function of all inputs yet hard-equivalent to the target function. \begin{figure}[t] \centering \includegraphics[width=1.0\textwidth]{margin-trick.png} - \caption{{\em todo}.} + \caption{{\em Margin packing for constructing gradient-rich, hard-equivalent functions}. A representative bit, $z$, is hard-equivalent to a discrete target function but gradient-sparse (e.g. $z=\operatorname{min}(x,y) \equiv x \wedge y$). On the left $z$ is low, $z<1/2$; on the right $z$ is high, $z>1/2$. We can pack a fraction of the margin between $z$ and the hard threshold $1/2$ with additional gradient-rich information without affecting hard-equivalence. A natural choice is the mean soft-bit, $\bar{\bf x} \in [0,1]$. The grey shaded areas denote the packed margins and the final augmented bit. On the left $\approx 60\%$ of the margin is packed; on the right $\approx 90\%$.} \label{fig:margin-trick} \end{figure} +% On the left, ${\bf x}=[0.9,0.23]$, $z=0.23$, $\bar{\bf x}=0.57$ and therefore $\approx 60\%$ of the margin is packed; on the right, ${\bf x}=[0.9,0.83]$, $z=0.83$, $\bar{\bf x}=0.87$, and therefore $\approx 90\%$ of the margin is packed. - -Define +Say we have a vector of soft-bit inputs ${\bf x}$ and the $i$th element represents the target discrete function (e.g. if our target is $x \wedge y$ then ${\bf x}=[x,y]$ and $i$ is 1 if $x 1/2 \\ -x_{i} + \operatorname{margin}({\bf x}, i) & \text{otherwise,} +1/2 + \operatorname{margin-fraction}({\bf x}, i) & \text{if } x_{i} > 1/2 \\ +x_{i} + \operatorname{margin-fraction}({\bf x}, i) & \text{otherwise.} \end{cases} \end{aligned} \end{equation*} +If the representative bit is high (resp. low) then the augmented bit is also high (resp. low). +But the augmented bit has a higher (resp. lower) value than the representative bit when below (resp. above) the $1/2$ threshold. The difference depends on the size of the available margin and the mean soft-bit value. Almost everywhere, an increase (resp. decrease) of the mean soft-bit increases (resp. decreases) the value of the augmented bit (see figure \ref{fig:margin-trick}). Note that if the $i$th bit is representative (i.e. hard-equivalent to the target function) then so is the augmented bit (see proposition \ref{prop:augmented}). We now use margin packing to define gradient-rich, hard-equivalents of basic boolean functions. +\subsection{Differentiable $\wedge$} -Define +We aim to construct a differentiable analogue of the boolean function $\bigwedge_{i=1}^{n} x_i$. A representative bit is $\operatorname{min}(x_{1},\dots,x_{n})$. The function \begin{equation*} \begin{aligned} -\partial\text{AND}: [0,1]^{n} &\to [0,1], \\ +\partial \wedge: [0,1]^{n} &\to [0,1], \\ {\bf x} &\mapsto \operatorname{augmented-bit}({\bf x}, \operatorname{argmin}\limits_{i} x[i]) \end{aligned} \end{equation*} -$\partial${AND} is hard-equivalent to the boolean function $\bigwedge_{i=1}^{n} x_i$ (proposition \ref{prop:and}). - +is therefore hard-equivalent to the boolean function $\bigwedge_{i=1}^{n} x_i$ (see proposition \ref{prop:and}). In the special case $n=2$ we get the piecewise function, \begin{equation*} -\partial\text{AND}(x, y) = +\partial\!\wedge\!(x, y) = \begin{cases} 1/2 + 1/2(x + y)(\operatorname{min}(x,y) - 1/2) & \text{if } \operatorname{min}(x,y) > 1/2 \\ \operatorname{min}(x,y) + 1/2(x + y)(1/2 - \operatorname{min}(x,y)) & \text{otherwise.} \end{cases} \end{equation*} - -$\partial${AND} is hard-equivalent to the boolean function $x \wedge y$ (proposition \ref{prop:and}). - - -We also want to maximise fan-in information where necessary (for backprop error signal) but also minimize it where necessary (e.g. picking a representative bit we want to optimize). Representative bit when we want a vector-wise update; otherwise, min/max. - -TODO: min/max are a degenerate case of sorting/ordering +Note that $\partial \wedge$ is differentiable almost everywhere and gradient-rich (see figure \ref{fig:gradient-rich}). \subsection{Differentiable $\vee$} -Define +The differentiable analogue of $\vee$ is identical to $\wedge$, except the representative bit is selected by $\operatorname{max}$. The function \begin{equation*} \begin{aligned} -\partial\text{OR}: [0,1]^{n} &\to [0,1], \\ +\partial\vee: [0,1]^{n} &\to [0,1], \\ {\bf x} &\mapsto \operatorname{augmented-bit}({\bf x}, \operatorname{argmax}\limits_{i} x[i]) \end{aligned} \end{equation*} -$\partial${OR} is hard-equivalent to the boolean function $\bigvee_{i=1}^{n} x_i$ (proposition \ref{prop:or}). +is hard-equivalent to the boolean function $\bigvee_{i=1}^{n} x_i$ (see proposition \ref{prop:or}). Note that $\partial \vee$ is differentiable almost everywhere and gradient-rich (see figure \ref{fig:gradient-rich}). +\begin{comment} Define \begin{equation*} \begin{aligned} -\partial\text{OR}(x, y) = +\partial\!\vee\!(x, y) = \begin{cases} 1/2 + 1/2(x + y)(\operatorname{max}(x,y) - 1/2) & \text{if } \operatorname{max}(x,y) > 1/2 \\ \operatorname{max}(x,y) + 1/2(x + y)(1/2 - \operatorname{max}(x,y)) & \text{otherwise.} @@ -207,7 +205,7 @@ \subsection{Differentiable $\vee$} \begin{comment} \begin{equation*} \begin{aligned} -\partial\text{OR}: [0,1]^{2} &\to [0,1], \\ +\partial \vee: [0,1]^{2} &\to [0,1], \\ (x, y) &\mapsto \begin{cases} 1/2 + 1/2(x + y)(m - 1/2) & \text{if } 2m > 1 \\ @@ -217,25 +215,30 @@ \subsection{Differentiable $\vee$} \end{equation*} \end{comment} -$\partial${OR} is hard-equivalent to the boolean function $x \vee y$ (proposition \ref{prop:or}). \subsection{Differentiable $\Rightarrow$} -Define +The differentiable analogue of $\Rightarrow$ is defined in terms of $\partial\vee$. The function \begin{equation*} \begin{aligned} -\partial\text{IMPLIES}: [0,1]^{2} &\to [0,1],\\ -(w, x) &\mapsto \partial\text{OR}(x, 1-w)\text{,} +\partial\!\Rightarrow: [0,1]^{2} &\to [0,1],\\ +(x, y) &\mapsto \partial\!\vee\!(y, 1-x)\text{,} \end{aligned} \end{equation*} -where $w$ is a weight and $x$ is a soft-bit value. -$\partial${IMPLIES} is hard-equivalent to the boolean function $w \Rightarrow x$ (proposition \ref{prop:implies}). +is hard-equivalent to $x \Rightarrow y$ (proposition \ref{prop:implies}). We can define analogues of all the basic boolean operators in a similar manner. \subsection{Differentiable majority} +Some boolean functions are particularly important for tractable learning. + +We also want to maximise fan-in information where necessary (for backprop error signal) but also minimize it where necessary (e.g. picking a representative bit we want to optimize). Representative bit when we want a vector-wise update; otherwise, min/max. + +TODO: min/max are a degenerate case of sorting/ordering + + \begin{figure}[t] \centering - \includegraphics[trim=0pt 0pt 0pt 0pt, clip, width=0.975\textwidth]{majority-gates.png} + \includegraphics[trim=0pt 0pt 0pt 0pt, clip, width=1.0\textwidth]{majority-gates.png} \caption{{\em Majority.}} \label{fig:majority-plot} \end{figure} @@ -315,36 +318,36 @@ \subsection{Logical layers} Define \begin{equation*} \begin{aligned} -\partial\text{NOT-LAYER}: [0,1]^{n \times m} \times [0,1]^{m} &\to [0,1]^{n \times m}, \\ +\partial \neg \text{LAYER}: [0,1]^{n \times m} \times [0,1]^{m} &\to [0,1]^{n \times m}, \\ ({\bf W}, {\bf x}) &\mapsto \begin{bmatrix} -\partial\text{NOT}(w_{1,1}, x_{1}) & \dots & \partial\text{NOT}(w_{1,m}, x_{m}) \\ +\partial \neg(w_{1,1}, x_{1}) & \dots & \partial \neg(w_{1,m}, x_{m}) \\ \vdots & \ddots & \vdots \\ -\partial\text{NOT}(w_{n,1}, x_{1}) & \dots & \partial\text{NOT}(w_{n,m}, x_{m}) +\partial \neg(w_{n,1}, x_{1}) & \dots & \partial \neg(w_{n,m}, x_{m}) \end{bmatrix} \end{aligned} \end{equation*} where ${\bf W}$ is a matrix of weights and ${\bf x}$ is a vector of soft-bits. -%[\partial\text{NOT}({\bf W}_{1}, {\bf x}), \dots, \partial\text{NOT}({\bf W}_{n}, {\bf x})] +%[\partial \neg({\bf W}_{1}, {\bf x}), \dots, \partial \neg({\bf W}_{n}, {\bf x})] Define \begin{equation*} \begin{aligned} -\partial\text{AND-NEURON}: [0,1]^{n} \times [0,1]^{n} &\to [0,1], \\ -({\bf w}, {\bf x}) &\mapsto \min(\partial\text{IMPLIES}(w_{1}, x_{1}), \dots, \partial\text{IMPLIES}(w_{n}, x_{n}))\text{,} +\partial\!\wedge\!\text{NEURON}: [0,1]^{n} \times [0,1]^{n} &\to [0,1], \\ +({\bf w}, {\bf x}) &\mapsto \min(\partial \Rightarrow(w_{1}, x_{1}), \dots, \partial \Rightarrow(w_{n}, x_{n}))\text{,} \end{aligned} \end{equation*} -where ${\bf w}$ is vector of weights and ${\bf x}$ is a vector of soft-bits. A single AND neuron maps $n$ soft-bit inputs to a single soft-bit value. A $\partial\text{AND-LAYER}$ of $n$ neurons maps $m$ soft-bit inputs to $n$ soft-bit outputs. +where ${\bf w}$ is vector of weights and ${\bf x}$ is a vector of soft-bits. A single AND neuron maps $n$ soft-bit inputs to a single soft-bit value. A $\partial\!\wedge\!\text{LAYER}$ of $n$ neurons maps $m$ soft-bit inputs to $n$ soft-bit outputs. Define \begin{equation*} \begin{aligned} -\partial\text{OR-NEURON}: [0,1]^{n} \times [0,1]^{n} &\to [0,1], \\ -({\bf w}, {\bf x}) &\mapsto \max(\partial\text{AND}(w_{1}, x_{1}), \dots, \partial\text{AND}(w_{n}, x_{n}))\text{.} +\partial\!\vee\!\text{NEURON}: [0,1]^{n} \times [0,1]^{n} &\to [0,1], \\ +({\bf w}, {\bf x}) &\mapsto \max(\partial \wedge(w_{1}, x_{1}), \dots, \partial \wedge(w_{n}, x_{n}))\text{.} \end{aligned} \end{equation*} -A single OR neuron maps $n$ soft-bit inputs to a single soft-bit value. A $\partial\text{OR-LAYER}$ of $n$ neurons maps $m$ soft-bit inputs to $n$ soft-bit outputs. +A single $\vee$ neuron maps $n$ soft-bit inputs to a single soft-bit value. A $\partial\!\vee\!\text{LAYER}$ of $n$ neurons maps $m$ soft-bit inputs to $n$ soft-bit outputs. \subsection{Architectures} @@ -364,6 +367,8 @@ \subsection{Architectures} \section{Experiments} +\subsection{Hardening} + \subsection{Binary Iris} \subsection{Noisy XOR} @@ -372,8 +377,8 @@ \subsection{Noisy XOR} \begin{figure}[t] \centering - \includegraphics[width=0.8\textwidth]{noisy-xor-architecture.png} - \caption{{\em A $\partial\mathbb{B}$ net for the noisy-xor problem}. The net concatenates the soft-bit input, ${\bf x}$ (length 12), with its negation, ${\bf 1 - x}$, and supplies the resulting vector (length 24) to a $\partial\text{AND-LAYER}$ (width 32), $\partial\text{OR-LAYER}$ (width 32), $\partial\text{NOT-LAYER}$ (width 16), and a final $\partial\text{MAJORITY}$ to produce a single soft-bit $y \in [0,1]$ (to predict odd parity) and its negation $1-y$ (to predict even parity). The net's weights, once hardened, consume $288$ bytes.} + \includegraphics[width=1.0\textwidth]{noisy-xor-architecture.png} + \caption{{\em A $\partial\mathbb{B}$ net for the noisy-xor problem}. The net concatenates the soft-bit input, ${\bf x}$ (length 12), with its negation, ${\bf 1 - x}$, and supplies the resulting vector (length 24) to a $\partial\!\wedge\!\text{LAYER}$ (width 32), $\partial\!\vee\!\text{LAYER}$ (width 32), $\partial \neg \text{LAYER}$ (width 16), and a final $\partial\text{MAJORITY}$ to produce a single soft-bit $y \in [0,1]$ (to predict odd parity) and its negation $1-y$ (to predict even parity). The net's weights, once hardened, consume $288$ bytes.} \label{fig:noisy-xor-architecture} \end{figure} @@ -421,14 +426,13 @@ \section*{Appendix} \section{Proofs} \begin{prop}\label{prop:not} - $\partial${NOT} is hard-equivalent to the boolean function - $\neg (x \oplus w)$. + $\partial \neg(x,y) \equiv \neg (x \oplus y)$. \begin{proof} Table \ref{not-table} is the truth table of the boolean function $\neg (x \oplus w)$, where $h(x) = \operatorname{harden}(x)$. \begin{table} \begin{center} \begin{tabular}{cccccc} - \multicolumn{1}{c}{$w$} &\multicolumn{1}{c}{$x$} &\multicolumn{1}{c}{$h(w)$} &\multicolumn{1}{c}{$h(x)$} &\multicolumn{1}{c}{$\partial\text{NOT}(h(w), h(x))$} &\multicolumn{1}{c}{$h(\partial\text{NOT}(h(w), h(x)))$} + \multicolumn{1}{c}{$x$} &\multicolumn{1}{c}{$y$} &\multicolumn{1}{c}{$h(x)$} &\multicolumn{1}{c}{$h(y)$} &\multicolumn{1}{c}{$\partial \neg(h(x), h(y))$} &\multicolumn{1}{c}{$h(\partial \neg(h(x), h(y)))$} \\ \hline \\ $\left[0, \frac{1}{2}\right]$ & $\left[0, \frac{1}{2}\right]$ & 0 & 0 & 1 & 1\\[0.1cm] $\left(\frac{1}{2}, 1\right]$ & $\left[0, \frac{1}{2}\right]$ &1 & 0 & 0 & 0\\[0.1cm] @@ -436,20 +440,27 @@ \section{Proofs} $\left(\frac{1}{2}, 1\right]$ & $\left(\frac{1}{2}, 1\right]$ &1 & 1 & 1 & 1\\[0.1cm] \end{tabular} \end{center} - \caption{$\partial${NOT} is hard-equivalent to $\neg (x \oplus w)$.}\label{not-table} + \caption{$\partial \neg(x,y) \equiv \neg (y \oplus x)$.}\label{not-table} \end{table} \end{proof} \end{prop} +\begin{prop}\label{prop:augmented} + If a representative bit is hard-equivalent to a target function then so is the augmented bit. + \begin{proof} + todo + \end{proof} +\end{prop} + \begin{prop}\label{prop:and} - $\partial${AND} is hard-equivalent to the boolean function $x \wedge y$. + $\partial\!\wedge\!(x,y) \equiv x \wedge y$. \begin{proof} Table \ref{and-table} is the truth table of the boolean function $x \wedge y$, where $h(x) = \operatorname{harden}(x)$.. \begin{table} \begin{center} \begin{tabular}{cccccc} - \multicolumn{1}{c}{$w$} &\multicolumn{1}{c}{$x$} &\multicolumn{1}{c}{$h(w)$} &\multicolumn{1}{c}{$h(x)$} &\multicolumn{1}{c}{$\partial\text{AND}(h(w), h(x))$} &\multicolumn{1}{c}{$h(\partial\text{AND}(h(w), h(x)))$} + \multicolumn{1}{c}{$x$} &\multicolumn{1}{c}{$y$} &\multicolumn{1}{c}{$h(x)$} &\multicolumn{1}{c}{$h(y)$} &\multicolumn{1}{c}{$\partial \wedge(h(x), h(y))$} &\multicolumn{1}{c}{$h(\partial \wedge(h(x), h(y)))$} \\ \hline \\ $\left[0, \frac{1}{2}\right]$ & $\left[0, \frac{1}{2}\right]$ & 0 & 0 & 0 & 0\\[0.1cm] $\left(\frac{1}{2}, 1\right]$ & $\left[0, \frac{1}{2}\right]$ &1 & 0 & $\frac{1}{4}$ & 0\\[0.1cm] @@ -457,19 +468,19 @@ \section{Proofs} $\left(\frac{1}{2}, 1\right]$ & $\left(\frac{1}{2}, 1\right]$ &1 & 1 & 1 & 1\\[0.1cm] \end{tabular} \end{center} - \caption{$\partial${AND} is hard-equivalent to $x \wedge y$.}\label{and-table} + \caption{$\partial \wedge(x,y) \equiv x \wedge y$.}\label{and-table} \end{table} \end{proof} \end{prop} \begin{prop}\label{prop:or} - $\partial${OR} is hard-equivalent to the boolean function $x \vee y$. + $\partial\!\vee\!(x,y) \equiv x \vee y$. \begin{proof} Table \ref{or-table} is the truth table of the boolean function $x \vee y$, where $h(x) = \operatorname{harden}(x)$.. \begin{table} \begin{center} \begin{tabular}{cccccc} - \multicolumn{1}{c}{$w$} &\multicolumn{1}{c}{$x$} &\multicolumn{1}{c}{$h(w)$} &\multicolumn{1}{c}{$h(x)$} &\multicolumn{1}{c}{$\partial\text{OR}(h(w), h(x))$} &\multicolumn{1}{c}{$h(\partial\text{OR}(h(w), h(x)))$} + \multicolumn{1}{c}{$x$} &\multicolumn{1}{c}{$y$} &\multicolumn{1}{c}{$h(x)$} &\multicolumn{1}{c}{$h(y)$} &\multicolumn{1}{c}{$\partial \vee(h(x), h(y))$} &\multicolumn{1}{c}{$h(\partial \vee(h(x), h(y)))$} \\ \hline \\ $\left[0, \frac{1}{2}\right]$ & $\left[0, \frac{1}{2}\right]$ & 0 & 0 & 0 & 0\\[0.1cm] $\left(\frac{1}{2}, 1\right]$ & $\left[0, \frac{1}{2}\right]$ &1 & 0 & $\frac{3}{4}$ & 1\\[0.1cm] @@ -477,19 +488,19 @@ \section{Proofs} $\left(\frac{1}{2}, 1\right]$ & $\left(\frac{1}{2}, 1\right]$ &1 & 1 & 1 & 1\\[0.1cm] \end{tabular} \end{center} - \caption{$\partial${OR} is hard-equivalent to $x \vee y$.}\label{or-table} + \caption{$\partial \vee(x,y) \equiv x \vee y$.}\label{or-table} \end{table} \end{proof} \end{prop} \begin{prop}\label{prop:implies} - $\partial${IMPLIES} is hard-equivalent to the boolean function $w \Rightarrow x$. + $\partial\!\Rightarrow\!(x,y) \equiv x \Rightarrow y$. \begin{proof} Table \ref{implies-table} is the truth table of the boolean function $x \Rightarrow y$, where $h(x) = \operatorname{harden}(x)$.. \begin{table} \begin{center} \begin{tabular}{cccccc} - \multicolumn{1}{c}{$w$} &\multicolumn{1}{c}{$x$} &\multicolumn{1}{c}{$h(w)$} &\multicolumn{1}{c}{$h(x)$} &\multicolumn{1}{c}{$\partial\text{IMPLIES}(h(w), h(x))$} &\multicolumn{1}{c}{$h(\partial\text{IMPLIES}(h(w), h(x)))$} + \multicolumn{1}{c}{$x$} &\multicolumn{1}{c}{$y$} &\multicolumn{1}{c}{$h(x)$} &\multicolumn{1}{c}{$h(y)$} &\multicolumn{1}{c}{$\partial \Rightarrow(h(x), h(y))$} &\multicolumn{1}{c}{$h(\partial \Rightarrow(h(x), h(y)))$} \\ \hline \\ $\left[0, \frac{1}{2}\right]$ & $\left[0, \frac{1}{2}\right]$ & 0 & 0 & $\frac{3}{4}$ & 1\\[0.1cm] $\left(\frac{1}{2}, 1\right]$ & $\left[0, \frac{1}{2}\right]$ &1 & 0 & 0 & 0\\[0.1cm] @@ -497,7 +508,7 @@ \section{Proofs} $\left(\frac{1}{2}, 1\right]$ & $\left(\frac{1}{2}, 1\right]$ &1 & 1 & $\frac{3}{4}$ & 1\\[0.1cm] \end{tabular} \end{center} - \caption{$\partial${IMPLIES} is hard-equivalent to $x \Rightarrow y$.}\label{implies-table} + \caption{$\partial \Rightarrow(x,y) \equiv x \Rightarrow y$.}\label{implies-table} \end{table} \end{proof} \end{prop} @@ -538,7 +549,7 @@ \section{Proofs} \begin{equation*} \begin{aligned} \partial\text{AND-LAYER}: [0,1]^{n \times m} \times [0,1]^{m} &\to [0,1]^{n}, \\ -({\bf W}, {\bf x}) &\mapsto [\partial\text{AND-NEURON}({\bf W}_{1}, {\bf x}), \dots, \partial\text{AND-NEURON}({\bf W}_{n}, {\bf x})] +({\bf W}, {\bf x}) 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"Output",ExpressionUUID->"22f9251a-c226-48d6-b7bf-c166c2a26e1a"] }, Open ]], Cell[CellGroupData[{ -Cell[4021827, 70571, 318, 7, 29, "Input",ExpressionUUID->"52628336-4c12-4de2-b09c-7f15f4d115a3"], -Cell[4022148, 70580, 259, 4, 96, "Output",ExpressionUUID->"c332ba01-b593-48f7-9440-c59ac5c44545"] +Cell[4021781, 70555, 318, 7, 29, "Input",ExpressionUUID->"52628336-4c12-4de2-b09c-7f15f4d115a3"], +Cell[4022102, 70564, 328, 5, 47, "Output",ExpressionUUID->"beeec97e-48a2-41a6-bf2f-02a116bed0f3"] }, Open ]] }, Open ]] }, Open ]] From 4b03307e52242325df99880b68223df0e3b6affe Mon Sep 17 00:00:00 2001 From: Ian Wright Date: Tue, 21 Mar 2023 18:21:58 +0000 Subject: [PATCH 048/113] more --- docs/db.tex | 66 +++++----- docs/proofs.nb | 326 ++++++++++++++++++++++++++++++++++++++++++------- 2 files changed, 312 insertions(+), 80 deletions(-) diff --git a/docs/db.tex b/docs/db.tex index 5e88b76..1672fc6 100644 --- a/docs/db.tex +++ b/docs/db.tex @@ -126,7 +126,7 @@ \subsection{Learning to negate} \subsection{Margin packing} -Say we aim to construct a differentiable analogue of $x \wedge y$. Note that $\operatorname{min}(x,y)$ essentially selects one of $x$ or $y$ as a `representative soft-bit' that is guaranteed hard-equivalent to $x \wedge y$. However, by selecting only one of $x$ or $y$ then $\operatorname{min}$ is also guaranteed to be gradient-sparse. We define a `margin packing' method to avoid this dilemma. +Say we aim to construct a differentiable analogue of $x \wedge y$. Note that $\operatorname{min}(x,y)$ essentially selects one of $x$ or $y$ as a representative soft-bit that is guaranteed hard-equivalent to $x \wedge y$. However, by selecting only one of $x$ or $y$ then $\operatorname{min}$ is also guaranteed to be gradient-sparse. We define a `margin packing' method to avoid this dilemma. The main idea of margin packing is (i) select a representative bit that is hard-equivalent to the target discrete function, and then (ii) pack a fraction of the margin between the representative bit and the hard threshold $1/2$ with gradient-rich information. The result is an augmented bit that is a function of all inputs yet hard-equivalent to the target function. @@ -138,14 +138,14 @@ \subsection{Margin packing} \end{figure} % On the left, ${\bf x}=[0.9,0.23]$, $z=0.23$, $\bar{\bf x}=0.57$ and therefore $\approx 60\%$ of the margin is packed; on the right, ${\bf x}=[0.9,0.83]$, $z=0.83$, $\bar{\bf x}=0.87$, and therefore $\approx 90\%$ of the margin is packed. -Say we have a vector of soft-bit inputs ${\bf x}$ and the $i$th element represents the target discrete function (e.g. if our target is $x \wedge y$ then ${\bf x}=[x,y]$ and $i$ is 1 if $x0} \to \mathbb{Z}_{> 0}$ as $n \mapsto 1 + \lfloor \frac{n-1}{2}\rfloor$. - -%Define $\operatorname{select}: [0,1]^n \times {1, 2, \ldots, n} \to [0,1]$ as $({\bf x}, i) \mapsto x_{i}$. - -%Define $\operatorname{majority-bit}: [0,1]^n \to [0,1]$ as ${\bf x} \mapsto \operatorname{select}( \operatorname{sort}({\bf x}), \operatorname{majority-index}(\lvert{\bf x}\rvert))$, where $\operatorname{sort}({\bf x})$ sorts the elements of ${\bf x}$ in ascending order. - -%Define $\operatorname{majority-delta}: [0,1]^n \to [0,1]$ as ${\bf x} \mapsto \operatorname{margin}(\operatorname{sort}({\bf x}), \operatorname{majority-index}({\bf x}))$. +$\operatorname{Maj}$ for $n$ bits in DNF form is a disjunction of $\binom{n}{k}$ conjunctive clauses of size $k$, where $k=\lceil n/2 \rceil$ and each clause checks if a unique combination of a majority of the $n$ bits are high; e.g. $\operatorname{Maj}(x, y, z) = (x \wedge y) \vee (x \wedge y) \vee (y \wedge z)$. Therefore, we could in principle implement a differentiable analogue of $\operatorname{Maj}$ in terms of $\partial\wedge$ and $\partial\vee$. However, the number of terms grows exponentially; e.g. $n=50$ generates over 100 trillion conjunctive clauses, which is infeasible. In circuit theory there is no known general algorithm for finding the minimal representation of $\operatorname{Maj}$ for arbitrary $n$. -Define +Instead, we observe that if $\operatorname{sort}({\bf x})$ sorts the elements of ${\bf x}$ in ascending order then the `middle' soft-bit is representative. For example, if ${\bf x} = [0.4, 0.9, 0.2]$ then $\operatorname{sort}({\bf x}) = [0.2, 0.4, 0.9]$ and the `middle' bit $x_{2}=0.4$ is low, which is hard-equivalent to $\operatorname{Maj}(0, 1, 0) = 0$. Define the index of the `middle' bit by \begin{equation*} \begin{aligned} - \partial\text{MAJORITY}: [0,1]^{n} &\to [0,1], \\ - {\bf x} &\mapsto \operatorname{augmented-bit}(\operatorname{sort}({\bf x}), \operatorname{majority-index}({\bf x})) +\operatorname{majority-index}: \mathbb{Z}_{>0} \to \mathbb{Z}_{> 0}\\ +n \mapsto \left\lceil \frac{n}{2}\right\rceil +\text{.} \end{aligned} \end{equation*} -\begin{comment} +Then the differentiable function \begin{equation*} \begin{aligned} -\partial\text{MAJORITY}: [0,1]^{n} &\to [0,1], \\ -{\bf x} &\mapsto - \begin{cases} - 1/2 + \delta & \text{if } m > 1/2 \\ - m + \delta & \text{otherwise,} - \end{cases} + \partial\!\operatorname{Maj}: [0,1]^{n} &\to [0,1], \\ + {\bf x} &\mapsto \operatorname{augmented-bit}(\operatorname{sort}({\bf x}), \operatorname{majority-index}({\bf x})) \end{aligned} \end{equation*} -\end{comment} -where ${\bf x}$ is a vector of soft-bits, and $\operatorname{sort}({\bf x})$ sorts the elements of ${\bf x}$ in ascending order. -$\partial${MAJORITY} is hard-equivalent to the boolean majority function (proposition \ref{prop:majority}). +is hard-equivalent to $\operatorname{Maj}({\bf x})$ (see proposition \ref{prop:majority}). Note that $\partial\!\operatorname{Maj}$ is differentiable almost everywhere and gradient-rich (see figure \ref{fig:majority-plot}). + -TODO: explain how we use sort to avoid space blow-up. TODO: discuss time-complexity of sort. +We also want to maximise fan-in information where necessary (for backprop error signal) but also minimize it where necessary (e.g. picking a representative bit we want to optimize). Representative bit when we want a vector-wise update; otherwise, min/max. + +TODO: min/max are a degenerate case of sorting/ordering + \subsection{Differentiable integer count} Define @@ -378,7 +376,7 @@ \subsection{Noisy XOR} \begin{figure}[t] \centering \includegraphics[width=1.0\textwidth]{noisy-xor-architecture.png} - \caption{{\em A $\partial\mathbb{B}$ net for the noisy-xor problem}. The net concatenates the soft-bit input, ${\bf x}$ (length 12), with its negation, ${\bf 1 - x}$, and supplies the resulting vector (length 24) to a $\partial\!\wedge\!\text{LAYER}$ (width 32), $\partial\!\vee\!\text{LAYER}$ (width 32), $\partial \neg \text{LAYER}$ (width 16), and a final $\partial\text{MAJORITY}$ to produce a single soft-bit $y \in [0,1]$ (to predict odd parity) and its negation $1-y$ (to predict even parity). The net's weights, once hardened, consume $288$ bytes.} + \caption{{\em A $\partial\mathbb{B}$ net for the noisy-xor problem}. The net concatenates the soft-bit input, ${\bf x}$ (length 12), with its negation, ${\bf 1 - x}$, and supplies the resulting vector (length 24) to a $\partial\!\wedge\!\text{LAYER}$ (width 32), $\partial\!\vee\!\text{LAYER}$ (width 32), $\partial \neg \text{LAYER}$ (width 16), and a final $\partial\!\operatorname{Maj}$ to produce a single soft-bit $y \in [0,1]$ (to predict odd parity) and its negation $1-y$ (to predict even parity). The net's weights, once hardened, consume $288$ bytes.} \label{fig:noisy-xor-architecture} \end{figure} diff --git a/docs/proofs.nb b/docs/proofs.nb index 5fbd22c..d5b0e18 100644 --- a/docs/proofs.nb +++ b/docs/proofs.nb @@ -10,10 +10,10 @@ NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] -NotebookDataLength[ 4039623, 70872] -NotebookOptionsPosition[ 4022470, 70574] -NotebookOutlinePosition[ 4023113, 70595] -CellTagsIndexPosition[ 4023070, 70592] +NotebookDataLength[ 4048234, 71106] +NotebookOptionsPosition[ 4029925, 70788] +NotebookOutlinePosition[ 4030568, 70809] +CellTagsIndexPosition[ 4030525, 70806] WindowFrame->Normal*) (* Beginning of Notebook Content *) @@ -70272,7 +70272,7 @@ Cell[BoxData[ 3.888327778900248*^9}, {3.888327859449986*^9, 3.8883278852434883`*^9}, { 3.888327982030218*^9, 3.8883280618627977`*^9}}, CellLabel-> - "In[1358]:=",ExpressionUUID->"8f1d6821-732d-4356-aecb-68dd42194a0c"], + 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Wright Date: Wed, 22 Mar 2023 11:39:24 +0000 Subject: [PATCH 049/113] more --- docs/db.tex | 21 +++++++-------------- docs/majority-gates.png | Bin 192547 -> 189478 bytes 2 files changed, 7 insertions(+), 14 deletions(-) diff --git a/docs/db.tex b/docs/db.tex index 1672fc6..4085894 100644 --- a/docs/db.tex +++ b/docs/db.tex @@ -231,7 +231,7 @@ \subsection{Differentiable majority} \begin{figure}[t] \centering \includegraphics[trim=0pt 0pt 0pt 0pt, clip, width=1.0\textwidth]{majority-gates.png} - \caption{{\em Majority.}} + \caption{{\em Differentiable boolean majority.} The boolean majority function for three variables in DNF form is $\operatorname{Maj}(x,y,z) = (x \wedge y) \vee (x \wedge y) \vee (y \wedge z)$. The upper row contains contour plots of $f(x,y,z) = \operatorname{min}(\operatorname{max}(x,y), \operatorname{max}(x,z), \operatorname{max}(y,z))$ for values of $z \in \{0.2, 0.4, 0.6, 0.8\}$. $f$ is differentiable and $\equiv \operatorname{Maj}$ but gradient-sparse (vertical and horizontal contours indicate constancy with respect to an input). Also, the number of terms in $f$ grows exponentially with the number of variables. The lower row contains contour plots of $\partial\!\operatorname{Maj}(x,y,z)$ for the same values of $z$. $\partial\!\operatorname{Maj}$ is differentiable and $\equiv \operatorname{Maj}$ yet gradient-rich (curved contours indicate variability with respect to any inputs). In addition, the number of terms in $\partial\!\operatorname{Maj}$ is constant with respect to the number of variables.} \label{fig:majority-plot} \end{figure} @@ -247,11 +247,11 @@ \subsection{Differentiable majority} \text{,} \end{aligned} \end{equation*} -Interpret each input bit $x_{i}$ as a vote, yes or no, for a binary decision. If the majority of voters are in favour then $\operatorname{Maj}$ outputs 1. The majority function, in the context of predictive model, aggregates multiple bits of weak evidence into a hard decision. Neural network binarization transforms real-valued activation functions into boolean majority \citep{10.5555/3157382.3157557}, which demonstrates their close association. We aim to construct a differentiable analogue of $\operatorname{Maj}$. +Interpret each input bit $x_{i}$ as a vote, yes or no, for a binary decision. If the majority of voters are in favour then $\operatorname{Maj}$ outputs 1. The majority function, in the context of predictive model, therefore aggregates multiple bits of weak evidence into a hard decision. Neural network binarization transforms real-valued activation functions into boolean threshold functions \citep{10.5555/3157382.3157557}, which indicates their close association. We aim to construct a differentiable analogue of $\operatorname{Maj}$. -$\operatorname{Maj}$ for $n$ bits in DNF form is a disjunction of $\binom{n}{k}$ conjunctive clauses of size $k$, where $k=\lceil n/2 \rceil$ and each clause checks if a unique combination of a majority of the $n$ bits are high; e.g. $\operatorname{Maj}(x, y, z) = (x \wedge y) \vee (x \wedge y) \vee (y \wedge z)$. Therefore, we could in principle implement a differentiable analogue of $\operatorname{Maj}$ in terms of $\partial\wedge$ and $\partial\vee$. However, the number of terms grows exponentially; e.g. $n=50$ generates over 100 trillion conjunctive clauses, which is infeasible. In circuit theory there is no known general algorithm for finding the minimal representation of $\operatorname{Maj}$ for arbitrary $n$. +$\operatorname{Maj}$ for $n$ bits in DNF form is a disjunction of $\binom{n}{k}$ conjunctive clauses of size $k$, where $k=\lceil n/2 \rceil$ and each clause checks if a unique combination of a majority of the $n$ bits are high; e.g. $\operatorname{Maj}(x, y, z) = (x \wedge y) \vee (x \wedge y) \vee (y \wedge z)$. Therefore, we could in principle implement a differentiable analogue of $\operatorname{Maj}$ in terms of $\partial\wedge$ and $\partial\vee$. However, the number of terms grows exponentially (e.g. $n=50$ generates over 100 trillion conjunctive clauses, which is infeasible) and there is no known general algorithm for finding the minimal representation of $\operatorname{Maj}$ for arbitrary $n$. -Instead, we observe that if $\operatorname{sort}({\bf x})$ sorts the elements of ${\bf x}$ in ascending order then the `middle' soft-bit is representative. For example, if ${\bf x} = [0.4, 0.9, 0.2]$ then $\operatorname{sort}({\bf x}) = [0.2, 0.4, 0.9]$ and the `middle' bit $x_{2}=0.4$ is low, which is hard-equivalent to $\operatorname{Maj}(0, 1, 0) = 0$. Define the index of the `middle' bit by +Instead, we trade-off time for memory costs. Observe that if the function $\operatorname{sort}({\bf x})$ sorts the elements of ${\bf x}$ in ascending order then the `middle' soft-bit is representative. For example, if ${\bf x} = [0.4, 0.9, 0.2]$ then $\operatorname{sort}({\bf x}) = [0.2, 0.4, 0.9]$ and the `middle' bit $x_{2}=0.4$ is low, which is hard-equivalent to $\operatorname{Maj}(0, 1, 0) = 0$. Define the index of the `middle' bit by \begin{equation*} \begin{aligned} \operatorname{majority-index}: \mathbb{Z}_{>0} \to \mathbb{Z}_{> 0}\\ @@ -259,21 +259,14 @@ \subsection{Differentiable majority} \text{.} \end{aligned} \end{equation*} -Then the differentiable function +Then, applying margin packing, define the differentiable function \begin{equation*} \begin{aligned} \partial\!\operatorname{Maj}: [0,1]^{n} &\to [0,1], \\ - {\bf x} &\mapsto \operatorname{augmented-bit}(\operatorname{sort}({\bf x}), \operatorname{majority-index}({\bf x})) + {\bf x} &\mapsto \operatorname{augmented-bit}(\operatorname{sort}({\bf x}), \operatorname{majority-index}({\bf x}))\text{,} \end{aligned} \end{equation*} -is hard-equivalent to $\operatorname{Maj}({\bf x})$ (see proposition \ref{prop:majority}). Note that $\partial\!\operatorname{Maj}$ is differentiable almost everywhere and gradient-rich (see figure \ref{fig:majority-plot}). - - -TODO: discuss time-complexity of sort. - -We also want to maximise fan-in information where necessary (for backprop error signal) but also minimize it where necessary (e.g. picking a representative bit we want to optimize). Representative bit when we want a vector-wise update; otherwise, min/max. - -TODO: min/max are a degenerate case of sorting/ordering +which is hard-equivalent to $\operatorname{Maj}({\bf x})$ (see proposition \ref{prop:majority}). Note that $\partial\!\operatorname{Maj}$ is differentiable almost everywhere and gradient-rich (see figure \ref{fig:majority-plot}). If $\operatorname{sort}$ is quicksort then the the average time-complexity of $\partial\!\operatorname{Maj}$ is $\mathcal{O}(n\log{}n)$, which is more expensive than $\partial\neg$, $\partial\wedge$, $\partial\vee$ and $\partial\!\Rightarrow$. \subsection{Differentiable integer count} diff --git a/docs/majority-gates.png b/docs/majority-gates.png index 8e72807d0c1996034a1c0640250536f1fafaa6f6..8554006f5ffa6b8c8668c890f341cc5f47c1e42e 100644 GIT binary patch delta 22920 zcmce;c{r8*yFR>{o}$s5Bqei(GDe2dQZmnD$ebZcGIKX6DoLgg%8+@kgrtN*$WX>e z=8Ty#d(ZXky?=Y}_xK&h``^2c&x`^Fmw6(T>C?&bd3$>NwR^ucWFF`Ae?Av%2D z6fRAbF>SskCC)S|&69QG+*+q~!sH>wb&1HpH7RGlC((n01KJ~Mwib4FQ8A%s$x+PA z$@9qrkFN7d6|4N5X)ZNxP0$M@vwz=_`c`3%O+sI_MqhH_T)~}d%Jdxi+t>)BvYPip ztlLW8%6%A8el_u8ZERlWV?+9wSYz!GdLx22{^HI>YnnWRt~|@L?8RPl={|EV%k$&g znFI|C3{;|o6^O=>8+mzGoa5t+c2Cx)yBe6|qG5QVJ#{J!fyax|G zw4WHLr2WB^;b@oOx5*N3yH&i6aC`M)uFu)$z=@{XWx9W_Czm5LFk`rh;I8`k@nd;8 z`-u~{UT0T%S67#6jJUOxm4}zt{BYyx;NB5`+hy_J5(|@kFO>@8m7;A##tuA;{P}cuj5uEpj 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