原英文檢視：http://neuralnetworksanddeeplearning.com/chap1.html

我們需要做的第⼀件事情是獲取 MNIST 資料。如果你是⼀個 git ⽤⼾，那麼你能夠

通過克隆這本書的程式碼倉庫獲得資料，
 實現我們的⽹絡來分類數字

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:])]
 

在這段程式碼中，列表 sizes 包含各層神經元的數量。例如，如果我們想建立⼀個在第⼀層有
2 個神經元，第⼆層有 3 個神經元，最後層有 1 個神經元的 Network 物件，我們應這樣寫程式碼：

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

Network 物件中的偏置和權重都是被隨機初始化的，使⽤ Numpy 的 np.random.randn 函式來⽣
成均值為 0，標準差為 1 的⾼斯分佈。這樣的隨機初始化給了我們的隨機梯度下降演算法⼀個起
點。在後⾯的章節中我們將會發現更好的初始化權重和偏置的⽅法，但是⽬前隨機地將其初始
化。注意 Network 初始化程式碼假設第⼀層神經元是⼀個輸⼊層，並對這些神經元不設定任何偏置，
因為偏置僅在後⾯的層中⽤於計算輸出。

有了這些，很容易寫出從⼀個 Network 例項計算輸出的程式碼。我們從定義 S 型函式開始：

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

注意，當輸⼊ z 是⼀個向量或者 Numpy 陣列時，Numpy ⾃動地按元素應⽤ sigmoid 函式，即
以向量形式。
我們然後對 Network 類新增⼀個 feedforward ⽅法，對於⽹絡給定⼀個輸⼊ a，返回對應的輸
出 6 。這個⽅法所做的是對每⼀層應⽤⽅程 (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
 

當然，我們想要 Network 物件做的主要事情是學習。為此我們給它們⼀個實現隨即梯度下降
演算法的 SGD ⽅法。程式碼如下。其中⼀些地⽅看似有⼀點神祕，我會在程式碼後⾯逐個分析

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 是⼀個 (x, y) 元組的列表，表⽰訓練輸⼊和其對應的期望輸出。變數 epochs 和
mini_batch_size 正如你預料的——迭代期數量，和取樣時的⼩批量資料的⼤⼩。 eta 是學習速率，
η。如果給出了可選引數 test_data ，那麼程式會在每個訓練器後評估⽹絡，並打印出部分進展。
這對於追蹤進度很有⽤，但相當拖慢執⾏速度。

在每個迭代期，它⾸先隨機地將訓練資料打亂，然後將它分成多個適當⼤
⼩的⼩批量資料。這是⼀個簡單的從訓練資料的隨機取樣⽅法。然後對於每⼀個 mini_batch
我們應⽤⼀次梯度下降。這是通過程式碼 self.update_mini_batch(mini_batch, eta) 完成的，它僅
僅使⽤ mini_batch 中的訓練資料，根據單次梯度下降的迭代更新⽹絡的權重和偏置。這是
update_mini_batch ⽅法的程式碼：

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)]

⼤部分⼯作由這⾏程式碼完成：

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

這⾏調⽤了⼀個稱為反向傳播的演算法，⼀種快速計算代價函式的梯度的⽅法。因此
update_mini_batch 的⼯作僅僅是對 mini_batch 中的每⼀個訓練樣本計算梯度，然後適當地更
新 self.weights 和 self.biases 。
我現在不會列出 self.backprop 的程式碼。我們將在下章中學習反向傳播是怎樣⼯作的，包括
self.backprop 的程式碼。現在，就假設它按照我們要求的⼯作，返回與訓練樣本 x 相關代價的適
當梯度

完整的程式

"""
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)