连载 | 深度学习入门第六讲

self.num_layers = len(sizes) self.sizes = sizes

self.biases = [np.random.randn(y, 1) for y in sizes[1:]] self.weights = [np.random.randn(y, x)

for x, y in zip(sizes[:-1], sizes[1:])]

"""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

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. If "test_data" is provided then the network will be evaluated against the test data after each

epoch, and partial progress printed out. This is useful for tracking progress, but slows things down substantially."""

if test_data: n_test = len(test_data) n = len(training_data)

for j in xrange(epochs): random.shuffle(training_data) mini_batches = [

training_data[k:k+mini_batch_size]for k in xrange(0, n, mini_batch_size)]

for mini_batch in mini_batches: self.update_mini_batch(mini_batch, eta) if test_data:

print "Epoch {0}: {1} / {2}".format( j, self.evaluate(test_data), n_test)

else: print "Epoch {0} complete".format(j)

gradient descent using backpropagation to a single mini batch. The "mini_batch" is a list of tuples "(x, y)", and "eta" is the learning rate."""

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)] self.weights = [w-(eta/len(mini_batch))*nw

for w, nw in zip(self.weights, nabla_w)] self.biases = [b-(eta/len(mini_batch))*nb

for b, nb in zip(self.biases, nabla_b)]

A module to implement the stochastic gradient descent learning

algorithm for a feedforward neural network. Gradients are calculated using backpropagation. 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 random

# Third-party libraries

class Network(object):

def __init__(self, sizes):"""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 a Gaussian

distribution with mean 0, and variance 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."""self.num_layers = len(sizes)

self.sizes = sizes self.biases = [np.random.randn(y, 1) for y in sizes[1:]]

self.weights = [np.random.randn(y, x) for x, y in zip(sizes[:-1], sizes[1:])]

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 adef SGD(self, training_data, epochs, mini_batch_size, eta,

test_data=None): """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. If ``test_data`` is provided then the network will be evaluated against the test data after each

epoch, and partial progress printed out. This is useful for tracking progress, but slows things down substantially."""

if test_data: n_test = len(test_data) n = len(training_data)for j in xrange(epochs):

random.shuffle(training_data) mini_batches = [

training_data[k:k+mini_batch_size]for k in xrange(0, n, mini_batch_size)]

for mini_batch in mini_batches: self.update_mini_batch(mini_batch, eta) if test_data:

print "Epoch {0}: {1} / {2}".format( j, self.evaluate(test_data), n_test)

else: print "Epoch {0} complete".format(j)

def update_mini_batch(self, mini_batch, eta):"""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)``, and ``eta``

is the learning rate.""" 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_w = [nw+dnw for nw, dnw in zip(nabla_w, delta_nabla_w)] self.weights = [w-(eta/len(mini_batch))*nw for w, nw in zip(self.weights, nabla_w)]

self.biases = [b-(eta/len(mini_batch))*nb for b, nb in zip(self.biases, nabla_b)]

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]# feedforwardactivation = x

activations = [x] # list to store all the activations, layer by layerzs = [] # 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 delta = self.cost_derivative(activations[-1], y) * \

sigmoid_prime(zs[-1]) nabla_b[-1] = delta 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 xrange(2, self.num_layers):

z = zs[-l] sp = sigmoid_prime(z)

delta = np.dot(self.weights[-l+1].transpose(), delta) * sp nabla_b[-l] = delta nabla_w[-l] = np.dot(delta, activations[-l-1].transpose())

return (nabla_b, nabla_w)

def evaluate(self, test_data):"""Return the number of test inputs for which the neural

network outputs the correct result. Note that the neural network's output is assumed to be the index of whichever neuron in the final layer has the highest activation."""

test_results = [(np.argmax(self.feedforward(x)), y) for (x, y) in test_data]

return sum(int(x == y) for (x, y) in test_results)def cost_derivative(self, output_activations, y):

"""Return the vector of partial derivatives \partial C_x / \partial a for the output activations."""

return (output_activations-y)

#### Miscellaneous functions def sigmoid(z):"""The sigmoid function."""

return 1.0/(1.0+np.exp(-z))

"""Derivative of the sigmoid function.""" return sigmoid(z)*(1-sigmoid(z))

... mnist_loader.load_data_wrapper()

Epoch 2: 9348 / 10000 ...

Epoch 27: 9528 / 10000 Epoch 28: 9542 / 10000

Epoch 29: 9534 / 10000

Epoch 2: 1135 / 10000 ...

Epoch 27: 2101 / 10000 Epoch 28: 2123 / 10000

Epoch 29: 2142 / 10000

>>> net.SGD(training_data, 30, 10, 100.0, test_data=test_data)

Epoch 1: 1009 / 10000 Epoch 2: 1009 / 10000

Epoch 3: 1009 / 10000 ... Epoch 27: 982 / 10000

Epoch 28: 982 / 10000 Epoch 29: 982 / 10000

mnist_loader ~~~~~~~~~~~~

A library to load the MNIST image data. For details of the data

structures that are returned, see the doc strings for ``load_data`` and ``load_data_wrapper``. In practice, ``load_data_wrapper`` is the function usually called by our neural network code.

"""

#### Libraries# Standard libraryimport cPickle

import gzip

# Third-party libraries import numpy as np

def load_data():"""Return the MNIST data as a tuple containing the training data,

the validation data, and the test data.

The ``training_data`` is returned as a tuple with two entries. The first entry contains the actual training images. This is a numpy ndarray with 50,000 entries. Each entry is, in turn, a

numpy ndarray with 784 values, representing the 28 * 28 = 784 pixels in a single MNIST image.

The second entry in the ``training_data`` tuple is a numpy ndarray

containing 50,000 entries. Those entries are just the digit values (0...9) for the corresponding images contained in the first entry of the tuple.

The ``validation_data`` and ``test_data`` are similar, except

each contains only 10,000 images. This is a nice data format, but for use in neural networks it's

helpful to modify the format of the ``training_data`` a little. That's done in the wrapper function ``load_data_wrapper()``, see

below. """

f = gzip.open('../data/mnist.pkl.gz', 'rb') training_data, validation_data, test_data = cPickle.load(f) f.close()

return (training_data, validation_data, test_data)

def load_data_wrapper():"""Return a tuple containing ``(training_data, validation_data, test_data)``. Based on ``load_data``, but the format is more

convenient for use in our implementation of neural networks.

In particular, ``training_data`` is a list containing 50,000 2-tuples ``(x, y)``. ``x`` is a 784-dimensional numpy.ndarray

containing the input image. ``y`` is a 10-dimensional numpy.ndarray representing the unit vector corresponding to the correct digit for ``x``.

``validation_data`` and ``test_data`` are lists containing 10,000

2-tuples ``(x, y)``. In each case, ``x`` is a 784-dimensional numpy.ndarry containing the input image, and ``y`` is the corresponding classification, i.e., the digit values (integers)

corresponding to ``x``.

Obviously, this means we're using slightly different formats for the training data and the validation / test data. These formats

turn out to be the most convenient for use in our neural network code."""tr_d, va_d, te_d = load_data()

training_inputs = [np.reshape(x, (784, 1)) for x in tr_d[0]] training_results = [vectorized_result(y) for y in tr_d[1]]

training_data = zip(training_inputs, training_results) validation_inputs = [np.reshape(x, (784, 1)) for x in va_d[0]] validation_data = zip(validation_inputs, va_d[1])

test_inputs = [np.reshape(x, (784, 1)) for x in te_d[0]] test_data = zip(test_inputs, te_d[1])

return (training_data, validation_data, test_data)

def vectorized_result(j):"""Return a 10-dimensional unit vector with a 1.0 in the jth 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] = 1.0return e