361 lines
15 KiB
Python
361 lines
15 KiB
Python
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"""network2.py
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~~~~~~~~~~~~~~
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An improved version of network.py, implementing the stochastic
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gradient descent learning algorithm for a feedforward neural network.
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Improvements include the addition of the cross-entropy cost function,
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regularization, and better initialization of network weights. Note
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that I have focused on making the code simple, easily readable, and
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easily modifiable. It is not optimized, and omits many desirable
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features.
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"""
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#### Libraries
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# Standard library
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import json
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import random
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import sys
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# Third-party libraries
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import numpy as np
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#### Define the quadratic and cross-entropy cost functions
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class QuadraticCost(object):
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@staticmethod
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def fn(a, y):
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"""Return the cost associated with an output ``a`` and desired output
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``y``.
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"""
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return 0.5*np.linalg.norm(a-y)**2
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@staticmethod
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def delta(z, a, y):
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"""Return the error delta from the output layer."""
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return (a-y) * sigmoid_prime(z)
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class CrossEntropyCost(object):
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@staticmethod
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def fn(a, y):
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"""Return the cost associated with an output ``a`` and desired output
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``y``. Note that np.nan_to_num is used to ensure numerical
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stability. In particular, if both ``a`` and ``y`` have a 1.0
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in the same slot, then the expression (1-y)*np.log(1-a)
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returns nan. The np.nan_to_num ensures that that is converted
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to the correct value (0.0).
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"""
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return np.sum(np.nan_to_num(-y*np.log(a)-(1-y)*np.log(1-a)))
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@staticmethod
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def delta(z, a, y):
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"""Return the error delta from the output layer. Note that the
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parameter ``z`` is not used by the method. It is included in
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the method's parameters in order to make the interface
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consistent with the delta method for other cost classes.
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"""
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return (a-y)
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#### Main Network class
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class Network(object):
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def __init__(self, sizes, cost=CrossEntropyCost):
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"""The list ``sizes`` contains the number of neurons in the respective
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layers of the network. For example, if the list was [2, 3, 1]
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then it would be a three-layer network, with the first layer
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containing 2 neurons, the second layer 3 neurons, and the
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third layer 1 neuron. The biases and weights for the network
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are initialized randomly, using
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``self.default_weight_initializer`` (see docstring for that
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method).
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"""
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self.num_layers = len(sizes)
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self.sizes = sizes
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self.default_weight_initializer()
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self.cost=cost
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def default_weight_initializer(self):
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"""Initialize each weight using a Gaussian distribution with mean 0
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and standard deviation 1 over the square root of the number of
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weights connecting to the same neuron. Initialize the biases
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using a Gaussian distribution with mean 0 and standard
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deviation 1.
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Note that the first layer is assumed to be an input layer, and
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by convention we won't set any biases for those neurons, since
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biases are only ever used in computing the outputs from later
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layers.
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"""
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self.biases = [np.random.randn(y, 1) for y in self.sizes[1:]]
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self.weights = [np.random.randn(y, x)/np.sqrt(x)
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for x, y in zip(self.sizes[:-1], self.sizes[1:])]
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def large_weight_initializer(self):
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"""Initialize the weights using a Gaussian distribution with mean 0
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and standard deviation 1. Initialize the biases using a
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Gaussian distribution with mean 0 and standard deviation 1.
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Note that the first layer is assumed to be an input layer, and
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by convention we won't set any biases for those neurons, since
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biases are only ever used in computing the outputs from later
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layers.
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This weight and bias initializer uses the same approach as in
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Chapter 1, and is included for purposes of comparison. It
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will usually be better to use the default weight initializer
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instead.
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"""
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self.biases = [np.random.randn(y, 1) for y in self.sizes[1:]]
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self.weights = [np.random.randn(y, x)
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for x, y in zip(self.sizes[:-1], self.sizes[1:])]
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def feedforward(self, a):
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"""Return the output of the network if ``a`` is input."""
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for b, w in zip(self.biases, self.weights):
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a = sigmoid(np.dot(w, a)+b)
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return a
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def SGD(self, training_data, epochs, mini_batch_size, eta,
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lmbda = 0.0,
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evaluation_data=None,
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monitor_evaluation_cost=False,
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monitor_evaluation_accuracy=False,
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monitor_training_cost=False,
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monitor_training_accuracy=False,
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early_stopping_n = 0):
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"""Train the neural network using mini-batch stochastic gradient
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descent. The ``training_data`` is a list of tuples ``(x, y)``
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representing the training inputs and the desired outputs. The
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other non-optional parameters are self-explanatory, as is the
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regularization parameter ``lmbda``. The method also accepts
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``evaluation_data``, usually either the validation or test
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data. We can monitor the cost and accuracy on either the
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evaluation data or the training data, by setting the
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appropriate flags. The method returns a tuple containing four
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lists: the (per-epoch) costs on the evaluation data, the
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accuracies on the evaluation data, the costs on the training
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data, and the accuracies on the training data. All values are
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evaluated at the end of each training epoch. So, for example,
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if we train for 30 epochs, then the first element of the tuple
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will be a 30-element list containing the cost on the
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evaluation data at the end of each epoch. Note that the lists
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are empty if the corresponding flag is not set.
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"""
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# early stopping functionality:
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best_accuracy=1
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training_data = list(training_data)
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n = len(training_data)
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if evaluation_data:
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evaluation_data = list(evaluation_data)
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n_data = len(evaluation_data)
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# early stopping functionality:
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best_accuracy=0
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no_accuracy_change=0
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evaluation_cost, evaluation_accuracy = [], []
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training_cost, training_accuracy = [], []
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for j in range(epochs):
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random.shuffle(training_data)
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mini_batches = [
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training_data[k:k+mini_batch_size]
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for k in range(0, n, mini_batch_size)]
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for mini_batch in mini_batches:
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self.update_mini_batch(
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mini_batch, eta, lmbda, len(training_data))
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print("Epoch %s training complete" % j)
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if monitor_training_cost:
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cost = self.total_cost(training_data, lmbda)
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training_cost.append(cost)
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print("Cost on training data: {}".format(cost))
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if monitor_training_accuracy:
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accuracy = self.accuracy(training_data, convert=True)
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training_accuracy.append(accuracy)
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print("Accuracy on training data: {} / {}".format(accuracy, n))
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if monitor_evaluation_cost:
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cost = self.total_cost(evaluation_data, lmbda, convert=True)
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evaluation_cost.append(cost)
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print("Cost on evaluation data: {}".format(cost))
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if monitor_evaluation_accuracy:
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accuracy = self.accuracy(evaluation_data)
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evaluation_accuracy.append(accuracy)
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print("Accuracy on evaluation data: {} / {}".format(self.accuracy(evaluation_data), n_data))
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# Early stopping:
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if early_stopping_n > 0:
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if accuracy > best_accuracy:
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best_accuracy = accuracy
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no_accuracy_change = 0
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#print("Early-stopping: Best so far {}".format(best_accuracy))
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else:
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no_accuracy_change += 1
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if (no_accuracy_change == early_stopping_n):
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#print("Early-stopping: No accuracy change in last epochs: {}".format(early_stopping_n))
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return evaluation_cost, evaluation_accuracy, training_cost, training_accuracy
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return evaluation_cost, evaluation_accuracy, \
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training_cost, training_accuracy
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def update_mini_batch(self, mini_batch, eta, lmbda, n):
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"""Update the network's weights and biases by applying gradient
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descent using backpropagation to a single mini batch. The
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``mini_batch`` is a list of tuples ``(x, y)``, ``eta`` is the
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learning rate, ``lmbda`` is the regularization parameter, and
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``n`` is the total size of the training data set.
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"""
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nabla_b = [np.zeros(b.shape) for b in self.biases]
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nabla_w = [np.zeros(w.shape) for w in self.weights]
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for x, y in mini_batch:
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delta_nabla_b, delta_nabla_w = self.backprop(x, y)
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nabla_b = [nb+dnb for nb, dnb in zip(nabla_b, delta_nabla_b)]
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nabla_w = [nw+dnw for nw, dnw in zip(nabla_w, delta_nabla_w)]
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self.weights = [(1-eta*(lmbda/n))*w-(eta/len(mini_batch))*nw
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for w, nw in zip(self.weights, nabla_w)]
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self.biases = [b-(eta/len(mini_batch))*nb
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for b, nb in zip(self.biases, nabla_b)]
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def backprop(self, x, y):
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"""Return a tuple ``(nabla_b, nabla_w)`` representing the
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gradient for the cost function C_x. ``nabla_b`` and
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``nabla_w`` are layer-by-layer lists of numpy arrays, similar
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to ``self.biases`` and ``self.weights``."""
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nabla_b = [np.zeros(b.shape) for b in self.biases]
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nabla_w = [np.zeros(w.shape) for w in self.weights]
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# feedforward
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activation = x
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activations = [x] # list to store all the activations, layer by layer
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zs = [] # list to store all the z vectors, layer by layer
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for b, w in zip(self.biases, self.weights):
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z = np.dot(w, activation)+b
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zs.append(z)
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activation = sigmoid(z)
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activations.append(activation)
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# backward pass
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delta = (self.cost).delta(zs[-1], activations[-1], y)
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nabla_b[-1] = delta
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nabla_w[-1] = np.dot(delta, activations[-2].transpose())
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# Note that the variable l in the loop below is used a little
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# differently to the notation in Chapter 2 of the book. Here,
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# l = 1 means the last layer of neurons, l = 2 is the
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# second-last layer, and so on. It's a renumbering of the
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# scheme in the book, used here to take advantage of the fact
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# that Python can use negative indices in lists.
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for l in range(2, self.num_layers):
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z = zs[-l]
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sp = sigmoid_prime(z)
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delta = np.dot(self.weights[-l+1].transpose(), delta) * sp
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nabla_b[-l] = delta
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nabla_w[-l] = np.dot(delta, activations[-l-1].transpose())
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return (nabla_b, nabla_w)
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def accuracy(self, data, convert=False):
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"""Return the number of inputs in ``data`` for which the neural
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network outputs the correct result. The neural network's
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output is assumed to be the index of whichever neuron in the
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final layer has the highest activation.
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The flag ``convert`` should be set to False if the data set is
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validation or test data (the usual case), and to True if the
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data set is the training data. The need for this flag arises
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due to differences in the way the results ``y`` are
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represented in the different data sets. In particular, it
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flags whether we need to convert between the different
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representations. It may seem strange to use different
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representations for the different data sets. Why not use the
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same representation for all three data sets? It's done for
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efficiency reasons -- the program usually evaluates the cost
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on the training data and the accuracy on other data sets.
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These are different types of computations, and using different
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representations speeds things up. More details on the
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representations can be found in
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mnist_loader.load_data_wrapper.
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"""
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if convert:
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results = [(np.argmax(self.feedforward(x)), np.argmax(y))
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for (x, y) in data]
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else:
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results = [(np.argmax(self.feedforward(x)), y)
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for (x, y) in data]
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result_accuracy = sum(int(x == y) for (x, y) in results)
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return result_accuracy
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def total_cost(self, data, lmbda, convert=False):
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"""Return the total cost for the data set ``data``. The flag
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``convert`` should be set to False if the data set is the
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training data (the usual case), and to True if the data set is
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the validation or test data. See comments on the similar (but
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reversed) convention for the ``accuracy`` method, above.
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"""
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cost = 0.0
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for x, y in data:
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a = self.feedforward(x)
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if convert: y = vectorized_result(y)
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cost += self.cost.fn(a, y)/len(data)
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cost += 0.5*(lmbda/len(data))*sum(np.linalg.norm(w)**2 for w in self.weights) # '**' - to the power of.
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return cost
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def save(self, filename):
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"""Save the neural network to the file ``filename``."""
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data = {"sizes": self.sizes,
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"weights": [w.tolist() for w in self.weights],
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"biases": [b.tolist() for b in self.biases],
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"cost": str(self.cost.__name__)}
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f = open(filename, "w")
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json.dump(data, f)
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f.close()
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#### Loading a Network
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def load(filename):
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"""Load a neural network from the file ``filename``. Returns an
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instance of Network.
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"""
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f = open(filename, "r")
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data = json.load(f)
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f.close()
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cost = getattr(sys.modules[__name__], data["cost"])
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net = Network(data["sizes"], cost=cost)
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net.weights = [np.array(w) for w in data["weights"]]
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net.biases = [np.array(b) for b in data["biases"]]
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return net
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#### Miscellaneous functions
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def vectorized_result(j):
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"""Return a 10-dimensional unit vector with a 1.0 in the j'th position
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and zeroes elsewhere. This is used to convert a digit (0...9)
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into a corresponding desired output from the neural network.
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"""
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e = np.zeros((10, 1))
|
||
|
|
e[j] = 1.0
|
||
|
|
return e
|
||
|
|
|
||
|
|
def sigmoid(z):
|
||
|
|
"""The sigmoid function."""
|
||
|
|
return 1.0/(1.0+np.exp(-z))
|
||
|
|
|
||
|
|
def sigmoid_prime(z):
|
||
|
|
"""Derivative of the sigmoid function."""
|
||
|
|
return sigmoid(z)*(1-sigmoid(z))
|