network2.py

"""network2.py
~~~~~~~~~~~~~~

An improved version of network.py, implementing the stochastic
gradient descent learning algorithm for a feedforward neural network.
Improvements include the addition of the cross-entropy cost function,
regularization, and better initialization of network weights.  Note
that I have focused on making the code simple, easily readable, and
easily modifiable.  It is not optimized, and omits many desirable
features.

"""

#### Libraries
# Standard library
import json
import random
import sys

# Third-party libraries
import numpy as np


#### Define the quadratic and cross-entropy cost functions

# cost for a single neuron where y and a are between 0 and 1.
class QuadraticCost(object):

    # used for determining total cost.
    @staticmethod
    def fn(a, y):
        """Return the cost associated with an output ``a`` and desired output
        ``y``.

        """
        # cost function as the MSE.
        return 0.5*np.linalg.norm(a-y)**2

    # used for cost of output layer and BP calculations.
    @staticmethod
    def delta(z, a, y):
        """Return the error delta from the output layer."""
        # BP1 in backpropagation.
        # derivative of cost quadratic cost above cross product.
        return (a-y) * sigmoid_prime(z)

# cost for a single neuron where y and a are between 0 and 1.
class CrossEntropyCost(object):

    # used for determining total cost.
    @staticmethod
    def fn(a, y):
        """Return the cost associated with an output ``a`` and desired output
        ``y``.  Note that np.nan_to_num is used to ensure numerical
        stability.  In particular, if both ``a`` and ``y`` have a 1.0
        in the same slot, then the expression (1-y)*np.log(1-a)
        returns nan.  The np.nan_to_num ensures that that is converted
        to the correct value (0.0).

        """
        # cost function calculated by integrating from cost gradient
        # for cross entropy.
        return np.sum(np.nan_to_num(-y*np.log(a)-(1-y)*np.log(1-a)))

    # used for cost of output layer and BP calculations.
    @staticmethod
    def delta(z, a, y):
        """Return the error delta from the output layer.  Note that the
        parameter ``z`` is not used by the method.  It is included in
        the method's parameters in order to make the interface
        consistent with the delta method for other cost classes.

        """
        # delta of actual and expected in the output layer.
        # this is the derivative of the cost function.
        return (a-y)


#### Main Network class
class Network(object):

    def __init__(self, sizes, cost=CrossEntropyCost):
        """The list ``sizes`` contains the number of neurons in the respective
        layers of the network.  For example, if the list was [2, 3, 1]
        then it would be a three-layer network, with the first layer
        containing 2 neurons, the second layer 3 neurons, and the
        third layer 1 neuron.  The biases and weights for the network
        are initialized randomly, using
        ``self.default_weight_initializer`` (see docstring for that
        method).

        """
        self.num_layers = len(sizes)
        self.sizes = sizes
        self.default_weight_initializer()
        self.cost=cost

    # this is the recommended initializer now, which has a tighter 
    # distribution of weights. 
    def default_weight_initializer(self):
        """Initialize each weight using a Gaussian distribution with mean 0
        and standard deviation 1 over the square root of the number of
        weights connecting to the same neuron.  Initialize the biases
        using a Gaussian distribution with mean 0 and standard
        deviation 1.

        Note that the first layer is assumed to be an input layer, and
        by convention we won't set any biases for those neurons, since
        biases are only ever used in computing the outputs from later
        layers.

        """
        # same as original or large weight initializer.
        self.biases = [np.random.randn(y, 1) for y in self.sizes[1:]]
        # basically reduce the sandard deviation by dividing by the sqrt of
        # the number of inputs into a given layer. this is akin to the 
        # regularization of weights in some ways. 
        self.weights = [np.random.randn(y, x)/np.sqrt(x)
                        for x, y in zip(self.sizes[:-1], self.sizes[1:])]

    def large_weight_initializer(self):
        """Initialize the weights using a Gaussian distribution with mean 0
        and standard deviation 1.  Initialize the biases using a
        Gaussian distribution with mean 0 and standard deviation 1.

        Note that the first layer is assumed to be an input layer, and
        by convention we won't set any biases for those neurons, since
        biases are only ever used in computing the outputs from later
        layers.

        This weight and bias initializer uses the same approach as in
        Chapter 1, and is included for purposes of comparison.  It
        will usually be better to use the default weight initializer
        instead.

        """
        # same as before
        self.biases = [np.random.randn(y, 1) for y in self.sizes[1:]]
        # same as before, allowing large weights to exist.
        self.weights = [np.random.randn(y, x)
                        for x, y in zip(self.sizes[:-1], self.sizes[1:])]

    # used for early stopping as well as...
    # output of network activations or calculations of accuracies.
    # output an n digit of numbers valued from 0 to 1 with on number greater 
    # than the rest to indicate its index is the predicted result of the network.
    def feedforward(self, a):
        """Return the output of the network if ``a`` is input."""
        for b, w in zip(self.biases, self.weights):
            a = sigmoid(np.dot(w, a)+b)
        return a

    # Stochastic Gradient Descent.
    def SGD(self, training_data, epochs, mini_batch_size, eta,
            lmbda = 0.0,
            evaluation_data=None,
            monitor_evaluation_cost=False,
            monitor_evaluation_accuracy=False,
            monitor_training_cost=False,
            monitor_training_accuracy=False,
            early_stopping_n = 0):
        """Train the neural network using mini-batch stochastic gradient
        descent.  The ``training_data`` is a list of tuples ``(x, y)``
        representing the training inputs and the desired outputs.  The
        other non-optional parameters are self-explanatory, as is the
        regularization parameter ``lmbda``.  The method also accepts
        ``evaluation_data``, usually either the validation or test
        data.  We can monitor the cost and accuracy on either the
        evaluation data or the training data, by setting the
        appropriate flags.  The method returns a tuple containing four
        lists: the (per-epoch) costs on the evaluation data, the
        accuracies on the evaluation data, the costs on the training
        data, and the accuracies on the training data.  All values are
        evaluated at the end of each training epoch.  So, for example,
        if we train for 30 epochs, then the first element of the tuple
        will be a 30-element list containing the cost on the
        evaluation data at the end of each epoch. Note that the lists
        are empty if the corresponding flag is not set.

        """

        # early stopping functionality:
        # best accuracy is 100%.
        best_accuracy=1

        # turn to list
        training_data = list(training_data)
        n = len(training_data)

        # same
        if evaluation_data:
            evaluation_data = list(evaluation_data)
            n_data = len(evaluation_data)

        # early stopping functionality:
        best_accuracy=0
        no_accuracy_change=0

        # initialize lists for tracking these values after each epoch until early stop.
        evaluation_cost, evaluation_accuracy = [], []
        training_cost, training_accuracy = [], []
        for j in range(epochs):
            random.shuffle(training_data)
            # create lists of size mini_batch_size
            mini_batches = [
                training_data[k:k+mini_batch_size]
                for k in range(0, n, mini_batch_size)]
            # run each minibatch (feedforward, cost calculation, back propagate).
            for mini_batch in mini_batches:
                self.update_mini_batch(
                    mini_batch, eta, lmbda, len(training_data))

            print("Epoch %s training complete" % j)

            if monitor_training_cost:
                cost = self.total_cost(training_data, lmbda)
                training_cost.append(cost)
                print("Cost on training data: {}".format(cost))
            if monitor_training_accuracy:
                accuracy = self.accuracy(training_data, convert=True)
                training_accuracy.append(accuracy)
                print("Accuracy on training data: {} / {}".format(accuracy, n))
            if monitor_evaluation_cost:
                cost = self.total_cost(evaluation_data, lmbda, convert=True)
                evaluation_cost.append(cost)
                print("Cost on evaluation data: {}".format(cost))
            if monitor_evaluation_accuracy:
                accuracy = self.accuracy(evaluation_data)
                evaluation_accuracy.append(accuracy)
                print("Accuracy on evaluation data: {} / {}".format(self.accuracy(evaluation_data), n_data))

            # Early stopping:
            if early_stopping_n > 0:
                if accuracy > best_accuracy:
                    best_accuracy = accuracy
                    no_accuracy_change = 0
                    #print("Early-stopping: Best so far {}".format(best_accuracy))
                else:
                    no_accuracy_change += 1

                if (no_accuracy_change == early_stopping_n):
                    #print("Early-stopping: No accuracy change in last epochs: {}".format(early_stopping_n))
                    return evaluation_cost, evaluation_accuracy, training_cost, training_accuracy

        return evaluation_cost, evaluation_accuracy, \
            training_cost, training_accuracy

    # similar to network.py but with L2 weight decay regularization.
    def update_mini_batch(self, mini_batch, eta, lmbda, n):
        """Update the network's weights and biases by applying gradient
        descent using backpropagation to a single mini batch.  The
        ``mini_batch`` is a list of tuples ``(x, y)``, ``eta`` is the
        learning rate, ``lmbda`` is the regularization parameter, and
        ``n`` is the total size of the training data set.

        """
        nabla_b = [np.zeros(b.shape) for b in self.biases]
        nabla_w = [np.zeros(w.shape) for w in self.weights]
        for x, y in mini_batch:
            delta_nabla_b, delta_nabla_w = self.backprop(x, y)
            nabla_b = [nb+dnb for nb, dnb in zip(nabla_b, delta_nabla_b)]
            nabla_w = [nw+dnw for nw, dnw in zip(nabla_w, delta_nabla_w)]
        # apply the L2 regularization parameter to the learning rules.
        # these decay the weights to prioritize smaller weights then subsequently
        # shift the weight as before according cost gradient.
        self.weights = [(1-eta*(lmbda/n))*w-(eta/len(mini_batch))*nw
                        for w, nw in zip(self.weights, nabla_w)]
        # biases doesn't change, as only weight is decayed.
        self.biases = [b-(eta/len(mini_batch))*nb
                       for b, nb in zip(self.biases, nabla_b)]

    # as in network.py, this actually includes feedforward then backpropagation.
    def backprop(self, x, y):
        """Return a tuple ``(nabla_b, nabla_w)`` representing the
        gradient for the cost function C_x.  ``nabla_b`` and
        ``nabla_w`` are layer-by-layer lists of numpy arrays, similar
        to ``self.biases`` and ``self.weights``."""
        nabla_b = [np.zeros(b.shape) for b in self.biases]
        nabla_w = [np.zeros(w.shape) for w in self.weights]

        # feedforward

        # calculate all activations in the network.
        activation = x
        activations = [x] # list to store all the activations, layer by layer
        zs = [] # list to store all the z vectors, layer by layer
        for b, w in zip(self.biases, self.weights):
            z = np.dot(w, activation)+b
            zs.append(z)
            activation = sigmoid(z)
            activations.append(activation)

        # backward pass

        # calculate the output layer error (aka delta) wrt activations.
        # depends on which cost function used. these are derivatives.
        delta = (self.cost).delta(zs[-1], activations[-1], y)
        # BP1
        nabla_b[-1] = delta
        # BP2
        nabla_w[-1] = np.dot(delta, activations[-2].transpose())
        # Note that the variable l in the loop below is used a little
        # differently to the notation in Chapter 2 of the book.  Here,
        # l = 1 means the last layer of neurons, l = 2 is the
        # second-last layer, and so on.  It's a renumbering of the
        # scheme in the book, used here to take advantage of the fact
        # that Python can use negative indices in lists.
        for l in range(2, self.num_layers):
            z = zs[-l]
            sp = sigmoid_prime(z)
            delta = np.dot(self.weights[-l+1].transpose(), delta) * sp
            # BP3
            nabla_b[-l] = delta
            # BP4
            nabla_w[-l] = np.dot(delta, activations[-l-1].transpose())
        return (nabla_b, nabla_w)

    # only in early stopping and in printing output (accuracy) of network.
    # percentage of correctly predicted.
    def accuracy(self, data, convert=False):
        """Return the number of inputs in ``data`` for which the neural
        network outputs the correct result. The neural network's
        output is assumed to be the index of whichever neuron in the
        final layer has the highest activation.

        The flag ``convert`` should be set to False if the data set is
        validation or test data (the usual case), and to True if the
        data set is the training data. The need for this flag arises
        due to differences in the way the results ``y`` are
        represented in the different data sets.  In particular, it
        flags whether we need to convert between the different
        representations.  It may seem strange to use different
        representations for the different data sets.  Why not use the
        same representation for all three data sets?  It's done for
        efficiency reasons -- the program usually evaluates the cost
        on the training data and the accuracy on other data sets.
        These are different types of computations, and using different
        representations speeds things up.  More details on the
        representations can be found in
        mnist_loader.load_data_wrapper.

        """
        # if data is training data.
        if convert:
            # store results as [(3,3),(3,4),etc....)]
            # here y is represented as an array and argmax finds index of largest value.
            results = [(np.argmax(self.feedforward(x)), np.argmax(y))
                       for (x, y) in data]
        # if validation or test data.
        else:
            # store results as [(3,3),(3,4),etc....)]
            # here y is represented as an integer.
            results = [(np.argmax(self.feedforward(x)), y)
                        for (x, y) in data]

        result_accuracy = sum(int(x == y) for (x, y) in results)
        return result_accuracy

    # average total cost of all the data sets. cost function + regularization cost.
    def total_cost(self, data, lmbda, convert=False):
        """Return the total cost for the data set ``data``.  The flag
        ``convert`` should be set to False if the data set is the
        training data (the usual case), and to True if the data set is
        the validation or test data.  See comments on the similar (but
        reversed) convention for the ``accuracy`` method, above.
        """
        cost = 0.0
        for x, y in data:
            # return result output of network
            a = self.feedforward(x)
            # return y in the form of a, vectors
            if convert: y = vectorized_result(y)
            # compute average cost using a combination of cost from cross-entropy/quadratic cost AND L2 regularization.
            # devide by len(data) along the way to get the average.
            # costs from cost functions
            cost += self.cost.fn(a, y)/len(data)
            # adding L2 regularization weight decay costs. (costs for having bigger weights).
            cost += 0.5*(lmbda/len(data))*sum(np.linalg.norm(w)**2 for w in self.weights) # '**' - to the power of.
        return cost

    # save the neural network as a json dump
    def save(self, filename):
        """Save the neural network to the file ``filename``."""
        data = {"sizes": self.sizes,
                "weights": [w.tolist() for w in self.weights],
                "biases": [b.tolist() for b in self.biases],
                "cost": str(self.cost.__name__)}
        f = open(filename, "w")
        json.dump(data, f)
        f.close()

#### Loading a Network from a json dump of weights, sizes, biases, cost.
def load(filename):
    """Load a neural network from the file ``filename``.  Returns an
    instance of Network.

    """
    f = open(filename, "r")
    data = json.load(f)
    f.close()
    cost = getattr(sys.modules[__name__], data["cost"])
    net = Network(data["sizes"], cost=cost)
    net.weights = [np.array(w) for w in data["weights"]]
    net.biases = [np.array(b) for b in data["biases"]]
    return net

#### Miscellaneous functions
def vectorized_result(j):
    """Return a 10-dimensional unit vector with a 1.0 in the j'th position
    and zeroes elsewhere.  This is used to convert a digit (0...9)
    into a corresponding desired output from the neural network.

    """
    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))