详解python实现识别手写MNIST数字集的程序_Python

我们需要做的第⼀件事情是获取 MNIST 数据。如果你是⼀个 git ⽤⼾，那么你能够通过克隆这本书的代码仓库获得数据，实现我们的⽹络来分类数字

1	`git clone https://github.com/mnielsen/neural-networks-and-deep-learning.git`

									class Network(object):

									def __init__(self, sizes):

									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:])]

在这段代码中，列表 sizes 包含各层神经元的数量。例如，如果我们想创建⼀个在第⼀层有2 个神经元，第⼆层有 3 个神经元，最后层有 1 个神经元的 Network 对象，我们应这样写代码：

1	`net` `=` `Network([2,` `3,` `1])`

Network 对象中的偏置和权重都是被随机初始化的，使⽤ Numpy 的 np.random.randn 函数来⽣成均值为 0，标准差为 1 的⾼斯分布。这样的随机初始化给了我们的随机梯度下降算法⼀个起点。在后⾯的章节中我们将会发现更好的初始化权重和偏置的⽅法，但是⽬前随机地将其初始化。注意 Network 初始化代码假设第⼀层神经元是⼀个输⼊层，并对这些神经元不设置任何偏置，因为偏置仅在后⾯的层中⽤于计算输出。有了这些，很容易写出从⼀个 Network 实例计算输出的代码。我们从定义 S 型函数开始：

1 2	`def` `sigmoid(z):` `return` `1.0/(1.0+np.exp(-z))`

注意，当输⼊ z 是⼀个向量或者 Numpy 数组时，Numpy ⾃动地按元素应⽤ sigmoid 函数，即以向量形式。

我们然后对 Network 类添加⼀个 feedforward ⽅法，对于⽹络给定⼀个输⼊ a，返回对应的输出 6 。这个⽅法所做的是对每⼀层应⽤⽅程 (22)：

									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

当然，我们想要 Network 对象做的主要事情是学习。为此我们给它们⼀个实现随即梯度下降算法的 SGD ⽅法。代码如下。其中⼀些地⽅看似有⼀点神秘，我会在代码后⾯逐个分析

									def 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)

training_data 是⼀个 (x, y) 元组的列表，表⽰训练输⼊和其对应的期望输出。变量 epochs 和mini_batch_size 正如你预料的——迭代期数量，和采样时的⼩批量数据的⼤⼩。 eta 是学习速率，η。如果给出了可选参数 test_data ，那么程序会在每个训练器后评估⽹络，并打印出部分进展。这对于追踪进度很有⽤，但相当拖慢执⾏速度。

在每个迭代期，它⾸先随机地将训练数据打乱，然后将它分成多个适当⼤⼩的⼩批量数据。这是⼀个简单的从训练数据的随机采样⽅法。然后对于每⼀个 mini_batch我们应⽤⼀次梯度下降。这是通过代码 self.update_mini_batch(mini_batch, eta) 完成的，它仅仅使⽤ mini_batch 中的训练数据，根据单次梯度下降的迭代更新⽹络的权重和偏置。这是update_mini_batch ⽅法的代码：

									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_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)]

⼤部分⼯作由这⾏代码完成：

1	`delta_nabla_b, delta_nabla_w` `=` `self.backprop(x, y)`

这⾏调⽤了⼀个称为反向传播的算法，⼀种快速计算代价函数的梯度的⽅法。因此update_mini_batch 的⼯作仅仅是对 mini_batch 中的每⼀个训练样本计算梯度，然后适当地更新 self.weights 和 self.biases 。我现在不会列出 self.backprop 的代码。我们将在下章中学习反向传播是怎样⼯作的，包括self.backprop 的代码。现在，就假设它按照我们要求的⼯作，返回与训练样本 x 相关代价的适当梯度

完整的程序

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									"""

									network.py

									~~~~~~~~~~

									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

									import numpy as np

									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 a

									 def 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_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)]

									 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

									  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

									  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))

									def sigmoid_prime(z):

									 """Derivative of the sigmoid function."""

									 return sigmoid(z)*(1-sigmoid(z))

									"""

									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 library

									import 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] = 1.0

									 return e

									# test network.py "cost function square func"

									import mnist_loader

									training_data, validation_data, test_data = mnist_loader.load_data_wrapper()

									import network

									net = network.Network([784, 10])

									net.SGD(training_data, 5, 10, 5.0, test_data=test_data)