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)]