手推反向传播+numpy实现

谁践踏了优雅 2023-10-06 14:09 27阅读 0赞

反向传播的重要性不必多说，手推也是必备基础，大厂面试要求用numpy实现一下BP也是经常的事。下面以一个简单的两层网络为例（简单到连bias都没有的那种），真·手推反向传播+numpy实现。

说明：

input：输入

net\_h：隐层

S（x）：sigmoid简写

SD（x）：sigmoid导数的简写

out\_h：隐层经过sigmoid激活的输出

net\_o：输出层

out\_o：经过经过sigmoid激活的输出层，也就是最后的输出

target（简写t1、t2）：Ground Truth

L：损失函数，一个简单的MSE

gw：L在w的偏导数

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3hpYW4wNzEwODMwMTE0_size_16_color_FFFFFF_t_70][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3hpYW4wNzEwODMwMTE0_size_16_color_FFFFFF_t_70 1][]

下面是代码：

1.对以上手推图片的实现：

import numpy as np
    
    def sigmoid(x):
        return 1 / (1 + np.exp(-x))
    def sigmoid_derivationx(y):
        return y * (1 - y)
    def back():
        input = np.array([0.9, 0.1], dtype='float')
        weight_1 = np.array([[0.15, 0.20], [0.25, 0.30]], dtype='float')
        weight_2 = np.array([[0.40, 0.45], [0.50, 0.55]], dtype='float')
        target_output = np.array([0.11, 0.89], dtype='float')
        learning_rate = 0.5
        steps = 1000
        while steps > 0:
            steps -= 1
            # 正向传播
            # layer-1
            net_h1, net_h2 = np.dot(input, weight_1)  # 第一层的计算结果
            out_h1, out_h2 = [1 / (1 + np.exp(-net_h1)), 1 / (1 + np.exp(-net_h2))]  # sigmoid激活后的输出
            # layer-2
            layer_2_input = np.array([out_h1, out_h2], dtype='float')
            net_o1, net_o2 = np.dot(layer_2_input, weight_2)  # 第二层的计算结果
            out_o1, out_o2 = [1 / (1 + np.exp(-net_o1)), 1 / (1 + np.exp(-net_o2))]  # sigmoid激活后的输出
            # 计算梯度,反向传播
            total_loss = sum([pow(target_output-real_output, 2)/2 for target_output, real_output in zip(target_output, [out_o1, out_o2])])
            print('网络的输出：step:%d loss=%f out_o1=%f out_o2=%f ' % (steps, total_loss, out_o1, out_o2))
            # gradient for layer-2
            gradient_w5 = -1 * (target_output[0] - out_o1) * out_o1 * (1 - out_o1) * out_h1
            gradient_w6 = -1 * (target_output[0] - out_o1) * out_o1 * (1 - out_o1) * out_h2
            gradient_w7 = -1 * (target_output[1] - out_o2) * out_o2 * (1 - out_o2) * out_h1
            gradient_w8 = -1 * (target_output[1] - out_o2) * out_o2 * (1 - out_o2) * out_h2
    
            # gradient for layer-1
            gradient_w1 = -1 * (target_output[0] - out_o1) * out_o1 * (1 - out_o1) * weight_2[0][0] * out_h1 * \
                          (1 - out_h1)*input[0] - (target_output[1] - out_o2) * out_o2 * (1 - out_o2) * weight_2[1][0] * \
                          out_h1 * (1 - out_h1)*input[0]
    
            gradient_w2 = -1 * (target_output[0] - out_o1) * out_o1 * (1 - out_o1) * weight_2[0][0] * out_h1 * \
                          (1 - out_h1) * input[1] - (target_output[1] - out_o2) * out_o2 * (1 - out_o2) * weight_2[1][0] * \
                          out_h1 * (1 - out_h1)*input[1]
    
            gradient_w3 = -1 * (target_output[0] - out_o1) * out_o1 * (1 - out_o1) * weight_2[0][1] * out_h2 * \
                          (1 - out_h2) * input[0] - (target_output[1] - out_o2) * out_o2 * (1 - out_o2) * weight_2[1][1] * \
                          out_h2 * (1 - out_h2)*input[0]
    
            gradient_w4 = -1 * (target_output[0] - out_o1) * out_o1 * (1 - out_o1) * weight_2[0][1] * out_h2 * \
                          (1 - out_h2) * input[1] - (target_output[1] - out_o2) * out_o2 * (1 - out_o2) * weight_2[1][1] * \
                          out_h2 * (1 - out_h2)*input[1]
    
            # update weight
            # layer -1
            weight_1[0][0] = weight_1[0][0] - learning_rate * gradient_w1
            weight_1[1][0] = weight_1[1][0] - learning_rate * gradient_w2
            weight_1[0][1] = weight_1[0][1] - learning_rate * gradient_w3
            weight_1[1][1] = weight_1[1][1] - learning_rate * gradient_w4
    back()

2.用矩阵相乘实现：

import numpy as np
    
    def sigmoid(x):
        return 1 / (1 + np.exp(-x))
    def sigmoid_derivationx(y):
        return y * (1 - y)
    def back_matrix():
        input = np.array([0.9, 0.1], dtype='float')
        weight_1 = np.array([[0.5, 0.5],
                             [0.5, 0.5]], dtype='float')
        weight_2 = np.array([[0.5, 0.5],
                             [0.5, 0.5]], dtype='float')
        target_output = np.array([0.11, 0.89], dtype='float')
        learning_rate = 0.5
        steps = 1000
        while steps > 0:
            steps -= 1
            # 正向传播
            # layer-1
            net_h = np.dot(input, weight_1)  # 第一层的计算结果
            out_h = sigmoid(net_h)  # sigmoid激活后的输出
            # layer-2
            layer_2_input = np.array(out_h, dtype='float')
            net_o = np.dot(layer_2_input, weight_2)  # 第二层的计算结果
            out_o = sigmoid(net_o)  # sigmoid激活后的输出
            # 计算梯度,反向传播
            total_loss = sum([pow(target_output-real_output, 2)/2 for target_output, real_output in zip(target_output, out_o)])
            print('网络的输出：step:%d loss=%f out_o1=%f out_o2=%f ' % (steps, total_loss, out_o[0], out_o[0]))
    
            gradient_weight_2 = np.zeros_like(weight_2)
            for i in range(gradient_weight_2.shape[1]):
                gradient_weight_2[:, i] = (out_o[i] - target_output[i]) * sigmoid_derivationx(out_o[i]) * out_h
    
            gradient_weight_1 = np.zeros_like(weight_1)
            for i in range(gradient_weight_1.shape[1]):
                dot1 = -1 * (target_output - out_o) * out_o * sigmoid_derivationx(out_h)
                dot2 = np.reshape(weight_2[:, i], (-1, 1))
                gradient_weight_1[:, i] = np.dot([dot1], dot2) * input
    
            # update weight
            # layer -1
            weight_1 = weight_1 - learning_rate * gradient_weight_1
    
            # layer-2
            weight_2 = weight_2 - learning_rate * gradient_weight_2
            if steps == 1:
                print('网络的输出：loss=%f out_o1=%f out_o2=%f ' % (total_loss, out_o[0], out_o[1]))
                break
    back_matrix()

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