动手写一个神经网络代码（附Backpropagation Algorithm代码分解）

先上Michal Daniel(传送门)的代码。类Network有六个成员函数，其中SGD、update_mini_batch、backprop负责计算每echo的残差、W和b偏导数、W和b的更新。feedforward、evaluation负责计算前向传导的值，可用于计算每echo训练集和验证集的error。cost_derivative计算网络最后一层的残差。

#### Libraries# Standard libraryimport random# Third-party librariesimport numpy as npclass 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:])]    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):        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), len(test_data))            else:                print "Epoch {0} complete".format(j)    def update_mini_batch(self, mini_batch, eta):        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):         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):  # feedforward 同时保存隐藏层计算的中间值结果            z = np.dot(w, activation)+b            zs.append(z)  # zs保存了每层神经元输入值            activation = sigmoid(z)            activations.append(activation)         delta = self.cost_derivative(activations[-1], y) * \            sigmoid_prime(zs[-1])        nabla_b[-1] = delta          nabla_w[-1] = np.dot(delta, activations[-2].transpose())           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()) # l 不是 1        return (nabla_b, nabla_w)    def evaluate(self, test_data)        test_results = [(np.argmax(self.feedforward(x)), y)                        for (x, y) in test_data]        # print test_results        return sum(int(x == y) for (x, y) in test_results)#cost的导数    def cost_derivative(self, output_activations, y):        return (output_activations-y)#### Miscellaneous functionsdef 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))

步骤分解：

首先需要传入的参数有层数、每层的神经元个数

根据传入参数初始化权重W和b，注意初始值必须是随机值，比如使用服从N(0,ϵ2)正态分布的随机值。如果初始化用全0，隐藏层会得到与输入值相同的函数，随机值目的是消除对称性。
输入数据X在每一epoch迭代前都要重新打乱，然后按照mini_batch_size大小切分数据，依次用每个batch训练更新W和b。每个epoch需要把所有batch训练完，训练完后可以测试下用现在的W和b能预测出什么样的结果来，并与真实值对比。然后进入下一epoch重复训练。

反馈传导步骤分解，公式代码可以对应：

1.进行前馈传导计算，利用前向传导公式，计算L1, L2, …直到Lnl的激活值。这个过程类似feedforward函数，不过我们需要保存隐藏层的计算结果以便后面求残差和偏导数。

z=sigmoid(w⋅x+b)

def backprop(self,x,y):    # 省略部分代码    activation = x     activations = [x] # list to store all the activations, layer by layer    zs = [] # list to strore all the z vaectors, layer by layer    for b, w in zip(self.biases, self.weights):        z = np.dot(w, activation)+b        zs.append(z)  # 保存了每层神经元输入值，后面        activation = sigmoid(z)        activations.append(activation)

z保存每层神经元输入值，activation保存每层神经元经过激活函数计算后的输出值

2.对输出层（nl层），残差就是激活值与实际值的差，计算：

δ(nl)=−(y−a(nl))⋅f′(z(nl))

def backprop(self,x,y):    # 省略部分代码    delta = self.cost_derivative(activations[-1], y) * \                sigmoid_prime(zs[-1])    # 求最后一层的残差    # nabla_b[-1] = delta      # nabla_w[-1] = np.dot(delta, activations[-2].transpose())def cost_derivative(self, output_activations, y):    return (output_activations-y)def sigmoid_prime(z):    """Derivative of the sigmoid function."""    return sigmoid(z)*(1-sigmoid(z))

3.对于l=nl−1,nl−2,...,2各层，计算残差，这步非常难理解，残差需要根据l+1层残差与l层W加权计算l层残差。给出公式如下：

δ(l)=((Wl)Tδl+1)⋅f′(z(l))

def backprop(self,x,y):    # 省略部分代码    # 代码里面 -l 表述倒数第 l 层。    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())

4.计算每层cost对w和b的偏导数

∇W(t)J(W,b;,x,y)=δ(l+1)(a(l))T

∇b(t)J(W,b;x,y)=δ(l+1)

 def backprop(self,x,y):    # 省略部分代码    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())

5.对于批量梯度下降法，样本从i=1到m，计算

ΔW(l):=ΔW(l)+∇W(t)J(W,b;x,y)

Δb(l):=Δb(l)+∇b(t)J(W,b;x,y)

def update_mini_batch(self, mini_batch, eta):    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)]

6.更新权重参数：

W(l)=W(l)−α[(1mΔW(l))]

b(l)=b(l)−α[1mΔb(l)]

def update_mini_batch(self, mini_batch, eta):    # 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)]

重复梯度下降法的迭代步骤来减小代价函数J(W,b)的值

改进方案

权重初始化改进：

W权重初始化从区间均匀随机取值,具体解释见http://blog.csdn.net/xbinworld/article/details/50603552 和http://neuralnetworksanddeeplearning.com/chap3.html#weight_initialization

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

增加正则化项

W(l)=W(l)−α[(1mΔW(l))+λW(l)]

def update_mini_batch(self, mini_batch, eta, lmbda, n):    """``lmbda`` is the regularization parameter, and        ``n`` is the total size of the training data set.    """    # 省略部分代码    self.weights = [(1-eta*(lmbda/n))*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)]

validation 求最优超参数

Quadratic Cost 二次损失函数