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 library
import random
# Third-party libraries
import numpy as np

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:])]
    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 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))

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**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**函数计算后的输出值

2.对输出层（nl层），残差就是**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