神经网络的前向传播和反向传播公式及代码实现

1.前向传播：

以三层神经网络为例，图中input为0.9，0.1，0.8，我们不用数字，用字母代替，设输入为x1,x2,x3,隐藏层的第一个神经元输出Ohidden1=S[(w11x1+b1)+(w21x2+b2)+(w31*x1+b3)](原文中未考虑偏置，这里我们加上偏置），S为激活函数sigmoid函数，其他神经元也是如此，用归纳的公式写出第l层第j个神经元的输出为：

也可以用矩阵表示：Xhidden = Whidden • I+bhidden,其中Whidden为输入层到隐藏层的权值矩阵，I为输入层的输入矩阵，因为是第一层所以不需要激活函数激活，b为偏置矩阵，X为隐藏层的输入矩阵，该层的输出需要经过激活函数，所以Ohidden=sigmoid(X),输出层的输入矩阵Xout=Wout• Ohidden+bout,Wout为隐藏层到输出层的权值矩阵，bout为隐藏层到输入层的偏置矩阵，最后的输出矩阵O=sigmoid(Xout)
2.反向传播

第一个输出节点的误差标记为e1，这个值等于由训练数据提供的所期望的输出值t1 与实际输出值o1 之间的差。也就是，e 1 = ( t 1-O 1)。

用矩阵来表示:

为了方便计算和好看我们使用简
单得多的e1 * w1,1 来代替e1 * w1,1/ ( w1,1 + w2,1)，所以：

即误差反向传播的误差权重矩阵为前向传播的转置矩阵，同理，有偏置时偏置反向矩阵为bT所以
ehidden=woutT *e+bT,当然这是简单的误差反向传播。实际中：我们用梯度下降法减少误差。下面是公式推导过程








α为学习率，调节变化的强度，确保不超调
用矩阵形式表示：

新偏置newb=b-αb
3.程序实现：
定义神经网络：

import numpy
import scipy.special
import matplotlib.pyplot
class neuralNetwork :
  def __init__(self, inputnodes, hiddennodes, outputnodes,learningrate) :
    self.inodes = inputnodes
    self.hnodes = hiddennodes
    self.onodes = outputnodes
    self.wih = numpy.random.normal(0.0, pow(self.hnodes, -0.5),(self.hnodes, self.inodes))
    self.who = numpy.random.normal(0.0, pow(self.onodes, -0.5),(self.onodes, self.hnodes))
    self.lr = learningrate
    self.activation_function = lambda x: scipy.special.expit(x)
    pass  
    #这是定义参数变量

  def train(self, inputs_list, targets_list) :
    inputs = numpy.array(inputs_list, ndmin=2).T
    targets = numpy.array(targets_list, ndmin=2).T
    hidden_inputs = numpy.dot(self.wih, inputs)
    hidden_outputs = self.activation_function(hidden_inputs)
    final_inputs = numpy.dot(self.who, hidden_outputs)
    final_outputs = self.activation_function(final_inputs)  #以上为前向传播
    output_errors = targets - final_outputs
    hidden_errors = numpy.dot(self.who.T, output_errors)
    self.who += self.lr * numpy.dot((output_errors *final_outputs * (1.0 - final_outputs)),numpy.transpose(hidden_outputs))
    self.wih += self.lr * numpy.dot((hidden_errors *hidden_outputs * (1.0 - hidden_outputs)), numpy.transpose(inputs))    #以上为反向传播权重更新
    pass

  def query(self, inputs_list) :
    inputs = numpy.array(inputs_list, ndmin=2).T
    hidden_inputs = numpy.dot(self.wih, inputs)
    hidden_outputs = self.activation_function(hidden_inputs)
    final_inputs = numpy.dot(self.who, hidden_outputs)
    final_outputs = self.activation_function(final_inputs)
    return final_outputs
    #这是前反向传播之后输入输出关系，即训练后输入输入值，返回输出值

以上是定义了基本的神经网络结构，我们用神经网络结构来训练和测试相关的训练集测试集，总体代码为下：

import numpy
import scipy.special
class neuralNetwork :
  def __init__(self, inputnodes, hiddennodes, outputnodes,learningrate) :
    self.inodes = inputnodes
    self.hnodes = hiddennodes
    self.onodes = outputnodes
    self.wih = numpy.random.normal(0.0, pow(self.hnodes, -0.5),(self.hnodes, self.inodes))
    self.who = numpy.random.normal(0.0, pow(self.onodes, -0.5),(self.onodes, self.hnodes))
    self.lr = learningrate
    self.activation_function = lambda x: scipy.special.expit(x)
    pass
  def train(self, inputs_list, targets_list) :
    inputs = numpy.array(inputs_list, ndmin=2).T
    targets = numpy.array(targets_list, ndmin=2).T
    hidden_inputs = numpy.dot(self.wih, inputs)
    hidden_outputs = self.activation_function(hidden_inputs)
    final_inputs = numpy.dot(self.who, hidden_outputs)
    final_outputs = self.activation_function(final_inputs)
    output_errors = targets - final_outputs
    hidden_errors = numpy.dot(self.who.T, output_errors)
    self.who += self.lr * numpy.dot((output_errors *final_outputs * (1.0 - final_outputs)),numpy.transpose(hidden_outputs))
    self.wih += self.lr * numpy.dot((hidden_errors *hidden_outputs * (1.0 - hidden_outputs)), numpy.transpose(inputs))
    pass
# query the neural network
  def query(self, inputs_list) :
    inputs = numpy.array(inputs_list, ndmin=2).T
    hidden_inputs = numpy.dot(self.wih, inputs)
    hidden_outputs = self.activation_function(hidden_inputs)
    final_inputs = numpy.dot(self.who, hidden_outputs)
    final_outputs = self.activation_function(final_inputs)
    return final_outputs


input_nodes=784
hidden_notes=200
output_nodes=10
learning_rate=0.1

n=neuralNetwork(input_nodes,hidden_notes,output_nodes,learning_rate)
training_data_file = open(r'C:\Users\尚鑫\Desktop\文件\mnist_train_100.csv','r',encoding='utf-8-sig')
training_data_list = training_data_file.readlines()
training_data_file.close()

epochs=5
for i in range(epochs):
    for record in training_data_list:
        all_values=record.split(',')
        inputs = (numpy.asfarray(all_values[1:]) / 255.0 * 0.99) +0.01
        targets = numpy.zeros(output_nodes) + 0.01
        targets[int(all_values[0])] = 0.99
        n.train(inputs, targets)
        pass
    pass
test_data_file = open(r'C:\Users\尚鑫\Desktop\文件\mnist_train_100.csv','r',encoding='utf-8-sig')
test_data_list = test_data_file.readlines()
test_data_file.close()

scorecard=[]
for record in test_data_list:
    all_values = record.split(',')
    correct_label = int(all_values[0])
    inputs = (numpy.asfarray(all_values[1:]) / 255.0 * 0.99) + 0.01
    outputs = n.query(inputs)
    label = numpy.argmax(outputs)
    if (label == correct_label):
        scorecard.append(1)
    else:
        scorecard.append(0)
        pass
    pass
scorecard_array = numpy.asarray(scorecard)
print ("performance = ", scorecard_array.sum()/scorecard_array.size)

最后输出的结果为：performance = 1.0