机器学习(11.4)--神经网络(nn)算法的深入与优化(4) -- CorssEntropyCost（交叉熵代价函数）数理分析与代码实现

这篇文章我们将从数理上对CorssEntropyCost进行讲解，同时附上实现的代码

可以参考

机器学习(11.3)--神经网络(nn)算法的深入与优化(3) -- QuadraticCost（二次方代价函数）数理分析

首先我们定义

因此在求得最后一层神经元


这时我们对最后一层的w,b求偏导数， 

我们在sigmoid函数定义为

由这个我们可以推出

其中x在程序代码对应的是a[-2],

def itemData(item,layers,weights,biases):
    '''单条记录的正反向计算'''
    #正向计算
    zs = []
    acts = [item[0]]
    for w,b in zip(weights,biases):
        z = np.dot(w,acts[-1]) + b
        zs.append(z)
        acts.append(sigmoid(z))
    
    #反向计算
    item_w = [np.zeros(b.shape) for b in weights]   
    item_b = [np.zeros(b.shape) for b in biases]  
    for index in range(-1,-1 * len(layers),-1):
        if index == -1:
            item_b[index] = acts[index] - item[1] 
        else:
            item_b[index] = np.dot(weights[index + 1].T,item_b[index + 1])
        #二次方代价函数 两个差别只是后面有没有乘 * sigmoid_deriv(zs[index])
        #在代码中的差异只是缩进不同，
        #item_b[index] = item_b[index] * sigmoid_deriv(zs[index]) 
            item_b[index] = item_b[index] * sigmoid_deriv(zs[index])  #交叉熵代价函数
        item_w[index] = np.dot(item_b[index],acts[index - 1].T)
    return item_w,item_b

虽然在这代码中的CorssEntropyCost（交叉熵代价函数） 与QuadraticCost（二次方代价函数）表现上只是一个缩进，

但原理却是千差万别，在使用QuadraticCost（二次方代价函数）时，使用不同激活函数影响不太大，

但如果使用CorssEntropyCost（交叉熵代价函数），因为中间有一个的过程

因此，得到a-y和激活函数是sigmoid是有相关的。

最后附上所有代码

# -*- coding:utf-8 -*-
import pickle  
import gzip  
import numpy as np  
import random

#激活函数
def sigmoid(z):  
    return 1.0 / (1.0 + np.exp(-z))  
  
def sigmoid_deriv(z):  
    return sigmoid(z) * (1 - sigmoid(z)) 

#读取数据
def loadData(trainingNum = None,testNum=None):
    with gzip.open(r'mnist.pkl.gz', 'rb')  as f:
        training_data, validation_data, test_data = pickle.load(f,encoding='bytes') 
    training_label = np.zeros([training_data[1].shape[0],10,1])
    for index,val in enumerate(training_data[1]): training_label[index][val] = 1
    training_data = list(zip(training_data[0].reshape(-1,784,1),training_label))
    test_data = list(zip(test_data[0].reshape(-1,784,1),test_data[1]))
    if trainingNum !=None:
        training_data = training_data[0:trainingNum]
    if trainingNum !=None:
        test_data = test_data[0:testNum] 

    return training_data,test_data

def batchData(batch,layers,weights,biases):
    batch_w = [np.zeros(b.shape) for b in weights]  
    batch_b = [np.zeros(b.shape) for b in biases]  
    for item in batch:
        item_w,item_b=itemData(item,layers,weights,biases)
        #当batch下每条记录计算完后加总
        for index in range(0,len(batch_w)):
            batch_w[index] = batch_w[index] + item_w[index]
            batch_b[index] = batch_b[index] + item_b[index]
    return batch_w,batch_b

def itemData(item,layers,weights,biases):
    '''单条记录的正反向计算'''
    #正向计算
    zs = []
    acts = [item[0]]
    for w,b in zip(weights,biases):
        z = np.dot(w,acts[-1]) + b
        zs.append(z)
        acts.append(sigmoid(z))
    
    #反向计算
    item_w = [np.zeros(b.shape) for b in weights]   
    item_b = [np.zeros(b.shape) for b in biases]  
    for index in range(-1,-1 * len(layers),-1):
        if index == -1:
            item_b[index] = acts[index] - item[1] 
        else:
            item_b[index] = np.dot(weights[index + 1].T,item_b[index + 1])
        #二次方代价函数 两个差别只是后面有没有乘 * sigmoid_deriv(zs[index])
        #在代码中的差异只是进位不同，
        #item_b[index] = item_b[index] * sigmoid_deriv(zs[index]) 
            item_b[index] = item_b[index] * sigmoid_deriv(zs[index])  #交叉熵代价函数
        item_w[index] = np.dot(item_b[index],acts[index - 1].T)
    return item_w,item_b

def predict(test_data,weights,biases):
    #6、正向计算测试集：计算出结果
    #7、和正确结果比较，统计出正确率
    correctNum=0
    for testImg,testLabel in test_data:
        for w,b in  zip( weights,biases):
            testImg= sigmoid(np.dot(w, testImg)+b)  
        if np.argmax(testImg)==testLabel : correctNum+=1
    return correctNum

def mnistNN(trainingNum = None,testNum = None,midLayes=[20,15],epochs=6,batchNum=10,learningRate=3):
    training_data,test_data=loadData(trainingNum,testNum)

    #1、读取数据，调整数据格式以适配算法，设置基本参数
    layers = [training_data[0][0].shape[0]]+midLayes+[training_data[0][1].shape[0]] 
    trainingNum = len(training_data)

    #2、建立初始化的weights和biases
    weights = [np.random.randn(layers[x + 1],layers[x])  for x in range(len(layers) - 1)]
    biases = [np.random.randn(layers[x + 1],1)  for x in range(len(layers) - 1)]

    for j in range(epochs):
        random.shuffle(training_data)  
        batchGroup = [training_data[x:x + batchNum] for x in range(0,trainingNum,batchNum)]
        for batch in batchGroup:
            batch_w,batch_b=batchData(batch,layers,weights,biases)
            #一组batch计算结束后，求平均并修正weights和biases
            for index in range(0,len(batch_w)):
                batch_w[index] = batch_w[index] / batchNum
                weights[index] = weights[index] - learningRate * batch_w[index]
                batch_b[index] = batch_b[index] / batchNum
                biases[index] = biases[index] - learningRate * batch_b[index]
        print("共 %d 轮训练，第 %d 轮训练结束，测试集数量为 %d 条，测试正确 %d 条。"%(epochs,j+1,len(test_data),predict(test_data,weights,biases)))


#参数组1  多调试几次，你会发现这组数据结果比较不稳定
mnistNN(midLayes=[30],epochs=15,learningRate=3)