反向传播、链式求导

今天药忘吃喽~ 2023-07-03 12:22 1阅读 0赞

# ![20191009191333910.png][] #

# [日萌社][Link 1] #

[人工智能AI：Keras PyTorch MXNet TensorFlow PaddlePaddle 深度学习实战（不定时更新）][AI_Keras PyTorch MXNet TensorFlow PaddlePaddle]

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[nltk 处理文本][nltk]

[注意力机制 SENet、CBAM][SENet_CBAM]

[卷积神经网络 处理文本：word2vec、TF-IDF、TextRank、字符卷积、词卷积、卷积神经网络文本分类模型的实现(Conv1D一维卷积、Conv2D二维卷积)][_word2vec_TF-IDF_TextRank_Conv1D_Conv2D]

[反向传播、链式求导][Link 2]

[梯度下降][Link 3]

[最小二乘法(LS算法)：实际为L2范数的一个具体应用(计算残差平方和)][LS_L2]

[线性回归 例子][Link 4]

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![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9henVzYS5ibG9nLmNzZG4ubmV0_size_16_color_FFFFFF_t_70][]![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9henVzYS5ibG9nLmNzZG4ubmV0_size_16_color_FFFFFF_t_70 1][]![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9henVzYS5ibG9nLmNzZG4ubmV0_size_16_color_FFFFFF_t_70 2][]![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9henVzYS5ibG9nLmNzZG4ubmV0_size_16_color_FFFFFF_t_70 3][]![20200206202933547.png][]![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9henVzYS5ibG9nLmNzZG4ubmV0_size_16_color_FFFFFF_t_70 4][]![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9henVzYS5ibG9nLmNzZG4ubmV0_size_16_color_FFFFFF_t_70 5][]![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9henVzYS5ibG9nLmNzZG4ubmV0_size_16_color_FFFFFF_t_70 6][]![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9henVzYS5ibG9nLmNzZG4ubmV0_size_16_color_FFFFFF_t_70 7][]![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9henVzYS5ibG9nLmNzZG4ubmV0_size_16_color_FFFFFF_t_70 8][]![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9henVzYS5ibG9nLmNzZG4ubmV0_size_16_color_FFFFFF_t_70 9][]![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9henVzYS5ibG9nLmNzZG4ubmV0_size_16_color_FFFFFF_t_70 10][]![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9henVzYS5ibG9nLmNzZG4ubmV0_size_16_color_FFFFFF_t_70 11][]![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9henVzYS5ibG9nLmNzZG4ubmV0_size_16_color_FFFFFF_t_70 12][]![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9henVzYS5ibG9nLmNzZG4ubmV0_size_16_color_FFFFFF_t_70 13][]![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9henVzYS5ibG9nLmNzZG4ubmV0_size_16_color_FFFFFF_t_70 14][]![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9henVzYS5ibG9nLmNzZG4ubmV0_size_16_color_FFFFFF_t_70 15][]![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9henVzYS5ibG9nLmNzZG4ubmV0_size_16_color_FFFFFF_t_70 16][]![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9henVzYS5ibG9nLmNzZG4ubmV0_size_16_color_FFFFFF_t_70 17][]![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9henVzYS5ibG9nLmNzZG4ubmV0_size_16_color_FFFFFF_t_70 18][]![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9henVzYS5ibG9nLmNzZG4ubmV0_size_16_color_FFFFFF_t_70 19][]![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9henVzYS5ibG9nLmNzZG4ubmV0_size_16_color_FFFFFF_t_70 20][]![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9henVzYS5ibG9nLmNzZG4ubmV0_size_16_color_FFFFFF_t_70 21][]![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9henVzYS5ibG9nLmNzZG4ubmV0_size_16_color_FFFFFF_t_70 22][]![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9henVzYS5ibG9nLmNzZG4ubmV0_size_16_color_FFFFFF_t_70 23][]![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9henVzYS5ibG9nLmNzZG4ubmV0_size_16_color_FFFFFF_t_70 24][]![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9henVzYS5ibG9nLmNzZG4ubmV0_size_16_color_FFFFFF_t_70 25][]![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9henVzYS5ibG9nLmNzZG4ubmV0_size_16_color_FFFFFF_t_70 26][]![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9henVzYS5ibG9nLmNzZG4ubmV0_size_16_color_FFFFFF_t_70 27][]![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9henVzYS5ibG9nLmNzZG4ubmV0_size_16_color_FFFFFF_t_70 28][]![20200206203111662.png][]

【程序3-3】开头部分
    import numpy as np
    import math
    import random
    
    def rand(a, b):
        return (b - a) * random.random() + a
    
    def make_matrix(m, n, fill=0.0):
        mat = []
        for i in range(m):
            mat.append([fill] * n)
        return mat
    
    def sigmoid(x):
        return 1.0 / (1.0 + math.exp(-x))
    
    def sigmod_derivate(x):
        return x * (1 - x)
    
    class BPNeuralNetwork:
        def __init__(self):
            self.input_n = 0
            self.hidden_n = 0
            self.output_n = 0
            self.input_cells = []
            self.hidden_cells = []
            self.output_cells = []
            self.input_weights = []
            self.output_weights = []
    
        def setup(self, ni, nh, no):
            self.input_n = ni + 1
            self.hidden_n = nh
            self.output_n = no
            self.input_cells = [1.0] * self.input_n
            self.hidden_cells = [1.0] * self.hidden_n
            self.output_cells = [1.0] * self.output_n
            self.input_weights = make_matrix(self.input_n, self.hidden_n)
            self.output_weights = make_matrix(self.hidden_n, self.output_n)
            # random activate
            for i in range(self.input_n):
                for h in range(self.hidden_n):
                    self.input_weights[i][h] = rand(-0.2, 0.2)
            for h in range(self.hidden_n):
                for o in range(self.output_n):
                    self.output_weights[h][o] = rand(-2.0, 2.0)
        # 前向传播
        def predict(self, inputs):
            for i in range(self.input_n - 1):
                self.input_cells[i] = inputs[i]
            for j in range(self.hidden_n):
                total = 0.0
                for i in range(self.input_n):
                    total += self.input_cells[i] * self.input_weights[i][j]
                self.hidden_cells[j] = sigmoid(total)
            for k in range(self.output_n):
                total = 0.0
                for j in range(self.hidden_n):
                    total += self.hidden_cells[j] * self.output_weights[j][k]
                self.output_cells[k] = sigmoid(total)
            return self.output_cells[:]
    
        # 反向传播
        def back_propagate(self, case, label, learn):
            self.predict(case) # 前向传播
    
            ### 计算输出层的误差 ###
            #先零初始化输出层中每个神经元的梯度值，实际为输出层的输出值对于误差的梯度值
            output_deltas = [0.0] * self.output_n 
            for k in range(self.output_n):
                #想要求出输出层的输出值对于误差的梯度值，首先要求出输出层每个神经元的误差，而输出层每个神经元的误差实际为真实值与模型预测值之间的差值。
                #即输出层每个神经元的误差值，实际为输出层中每个神经元的输出值和每个真实值之间的差值。
                error = label[k] - self.output_cells[k] 
                #首先把输出层该神经元中已经经过了激活函数的输出值先通过sigmod导函数求导，然后乘以该神经元的误差值，最终得出该神经元的梯度值
                output_deltas[k] = sigmod_derivate(self.output_cells[k]) * error 
    
            ### 计算隐藏层(全连接层)的误差 ###
            #先零初始化隐藏层(全连接层)中每个神经元的梯度值，实际为隐藏层的输出值对于误差的梯度值
            hidden_deltas = [0.0] * self.hidden_n 
            for j in range(self.hidden_n):
                #想要求出隐藏层的输出值对于误差的梯度值，首先要求出隐藏层每个神经元的误差，但是对于隐藏层，由于缺少直接的目标值来计算隐藏层中每个神经元的误差，
                #因此需要通过间接的方式计算隐藏层的误差项，首先通过下一层输出层已知的每个神经元的梯度值乘以隐藏层中二维权重矩阵中第j个神经元的output_n个权重值，
                #即加权求和得出隐藏层中第j个神经元的误差。
                error = 0.0
                for k in range(self.output_n):
                    #通过下一层输出层已知的每个神经元的梯度值乘以隐藏层中二维权重矩阵中第j个神经元的output_n个权重值，即加权求和得出隐藏层中第j个神经元的误差
                    error += output_deltas[k] * self.output_weights[j][k] 
                #首先把隐藏层该神经元中已经经过了激活函数的输出值先通过sigmod导函数求导，然后乘以该神经元的误差值，最终得出该神经元的梯度值
                hidden_deltas[j] = sigmod_derivate(self.hidden_cells[j]) * error
    
            ### 更新输出层权重 ###
            for j in range(self.hidden_n):
                for k in range(self.output_n):
                    #θ-= lr * 输出层的梯度值 * 输出层的输入
                    self.output_weights[j][k] += learn * output_deltas[k] * self.hidden_cells[j]
    
            ### 更新隐藏层权重 ###
            for i in range(self.input_n):
                for j in range(self.hidden_n):
                    #θ-= lr * 隐藏层的梯度值 * 隐藏层的输入
                    self.input_weights[i][j] += learn * hidden_deltas[j] * self.input_cells[i]
    
            error = 0
            for o in range(len(label)):
                error += 0.5 * (label[o] - self.output_cells[o]) ** 2 #计算均方误差
            return error
    
        def train(self, cases, labels, limit=100, learn=0.05):
            for i in range(limit):
                error = 0
                for i in range(len(cases)):
                    label = labels[i]
                    case = cases[i]
                    error += self.back_propagate(case, label, learn)
            pass
    
        def test(self):
            cases = [
                [0, 0],
                [0, 1],
                [1, 0],
                [1, 1],
            ]
            labels = [[0], [1], [1], [0]]
            self.setup(2, 5, 1)
            self.train(cases, labels, 10000, 0.05)
            for case in cases:
                print(self.predict(case))
    
    if __name__ == '__main__':
        nn = BPNeuralNetwork()
        nn.test()

--------------------

# **计算每一层(输出层/隐藏层)的误差和梯度**![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9henVzYS5ibG9nLmNzZG4ubmV0_size_16_color_FFFFFF_t_70 29][] #

### 计算输出层的误差 ###
            #先零初始化输出层中每个神经元的梯度值，实际为输出层的输出值对于误差的梯度值
            output_deltas = [0.0] * self.output_n 
            for k in range(self.output_n):
                #想要求出输出层的输出值对于误差的梯度值，首先要求出输出层每个神经元的误差，而输出层每个神经元的误差实际为真实值与模型预测值之间的差值。
                #即输出层每个神经元的误差值，实际为输出层中每个神经元的输出值和每个真实值之间的差值。
                error = label[k] - self.output_cells[k] 
                #首先把输出层该神经元中已经经过了激活函数的输出值先通过sigmod导函数求导，然后乘以该神经元的误差值，最终得出该神经元的梯度值
                output_deltas[k] = sigmod_derivate(self.output_cells[k]) * error 
    
            ### 计算隐藏层(全连接层)的误差 ###
            #先零初始化隐藏层(全连接层)中每个神经元的梯度值，实际为隐藏层的输出值对于误差的梯度值
            hidden_deltas = [0.0] * self.hidden_n 
            for j in range(self.hidden_n):
                #想要求出隐藏层的输出值对于误差的梯度值，首先要求出隐藏层每个神经元的误差，但是对于隐藏层，由于缺少直接的目标值来计算隐藏层中每个神经元的误差，
                #因此需要通过间接的方式计算隐藏层的误差项，首先通过下一层输出层已知的每个神经元的梯度值乘以隐藏层中二维权重矩阵中第j个神经元的output_n个权重值，
                #即加权求和得出隐藏层中第j个神经元的误差。
                error = 0.0
                for k in range(self.output_n):
                    #通过下一层输出层已知的每个神经元的梯度值乘以隐藏层中二维权重矩阵中第j个神经元的output_n个权重值，即加权求和得出隐藏层中第j个神经元的误差
                    error += output_deltas[k] * self.output_weights[j][k] 
                #首先把隐藏层该神经元中已经经过了激活函数的输出值先通过sigmod导函数求导，然后乘以该神经元的误差值，最终得出该神经元的梯度值
                hidden_deltas[j] = sigmod_derivate(self.hidden_cells[j]) * error

--------------------

# **每一层的权重更新 ** #

### 更新输出层权重 ###
            for j in range(self.hidden_n):
                for k in range(self.output_n):
                    #θ-= lr * 输出层的梯度值 * 输出层的输入
                    self.output_weights[j][k] += learn * output_deltas[k] * self.hidden_cells[j]
    
            ### 更新隐藏层权重 ###
            for i in range(self.input_n):
                for j in range(self.hidden_n):
                    #θ-= lr * 隐藏层的梯度值 * 隐藏层的输入
                    self.input_weights[i][j] += learn * hidden_deltas[j] * self.input_cells[i]

[20191009191333910.png]: https://img-blog.csdnimg.cn/20191009191333910.png
[Link 1]: http://www.rimengshe.com/
[AI_Keras PyTorch MXNet TensorFlow PaddlePaddle]: https://blog.csdn.net/zimiao552147572/article/details/88867161
[nltk]: https://azusa.blog.csdn.net/article/details/104203185
[SENet_CBAM]: https://azusa.blog.csdn.net/article/details/104201845
[_word2vec_TF-IDF_TextRank_Conv1D_Conv2D]: https://azusa.blog.csdn.net/article/details/104208463
[Link 2]: https://azusa.blog.csdn.net/article/details/104201797
[Link 3]: https://azusa.blog.csdn.net/article/details/104201755
[LS_L2]: https://azusa.blog.csdn.net/article/details/104201735
[Link 4]: https://azusa.blog.csdn.net/article/details/104201717
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