贪玩ML系列之一个BP玩一天

手写串行BP算法，可调batch_size

既要：1、输入层f(x)=x  隐藏层sigmoid 输出层f(x)=x

2、run函数实现单条数据的一次前馈

3、train函数读入所有数据for循环处理每条数据。

循环中：

首先调用run函数，得到各层的值

self.input_nodes_value

self.hidden_nodes_value

self.output_nodes_value

然后计算输出层误差和delta

4、关键函数：用于前馈的sigmoid和用于反馈的sigmoid的导数

 self.activation_function = lambda x : 1/(1+np.exp(-x))  # sigmoid函数，用于正向传播
 self.delta_activation_function = lambda x: x-x**2 # sigmoid一阶导，用于反向传播

5、反向传播

使用梯度下降方法

下面是推导隐藏层（实际上为relu层）到输出层的权重w[h][o]的梯度下降公式的过程，对应的几个变量在下面的代码中用红色标出

关于梯度下降公式推导：

https://blog.csdn.net/wfei101/article/details/80807749

https://www.jianshu.com/p/17191c57d7e9

batch_size=1

# 输入层没有激活函数f(x)=x，隐藏层激活函数sigmoid，输出层激活函数f(x)=x
class NeuralNetwork(object):
    def __init__(self, input_nodes, hidden_nodes, output_nodes, learning_rate):

        # 各层节点个数
        self.input_nodes = input_nodes
        self.hidden_nodes = hidden_nodes
        self.output_nodes = output_nodes

#         创建三个一维数组存放三层节点的值
#         print(str(self.input_nodes)+" "+str(self.hidden_nodes)+" "+str(self.output_nodes))
        self.input_nodes_value=[0.0]*input_nodes
        self.hidden_nodes_value=[0.0]*hidden_nodes
        self.output_nodes_value=[0.0]*output_nodes

        # Initialize weights
        self.weights_input_to_hidden = np.random.normal(0.0, self.input_nodes**-0.5, (self.input_nodes, self.hidden_nodes))#输入层>>隐藏层权重矩阵

        self.weights_hidden_to_output = np.random.normal(0.0, self.hidden_nodes**-0.5, (self.hidden_nodes, self.output_nodes))#隐藏层>>输出层权重矩阵

        self.learning_rate = learning_rate#学习率

        self.activation_function = lambda x : 1/(1+np.exp(-x))  # sigmoid函数，用于正向传播
        self.delta_activation_function = lambda x: x-x**2 # sigmoid一阶导，用于反向传播

        self.change_to_fix_weights_h2o=[[0.0]*self.output_nodes]*self.hidden_nodes#存储隐藏层>>输出层权重调整量
        self.change_to_fix_weights_i2h=[[0.0]*self.hidden_nodes]*self.input_nodes#存储输入层>>隐藏层权重调整量
#         print("xxxx")
#         print(self.change_to_fix_weights_h2o)
#         print(self.change_to_fix_weights_i2h)

    def train(self, features, targets):#完成n条数据的一次前向传递和反向传递，每个batch调整一次权重矩阵
        '''
            features: 2D array, each row is one data record, each column is a feature
            targets: 1D array of target values

        '''
        n=features.shape[0]#数据条数
#         print(features)
#         print(targets)

        counter=batch_size
        for ii in range(0,n):

            self.run(features[ii])#调用前向传播

            print(self.output_nodes_value)

            error_o=[0.0]*self.output_nodes#输出层误差
            error_h=[0.0]*self.hidden_nodes#隐藏层误差
            output_deltas=[0.0]*self.output_nodes
            hidden_deltas=[0.0]*self.hidden_nodes

            for o in range(self.output_nodes): # 输 出 层
                error_o[o]=targets[ii][o]-self.output_nodes_value[o]#计算输出层误差
#                 output_deltas[o]=self.delta_activation_function(self.output_nodes_value[o])*error_o[o]#输出层反向传播（求导）
                output_deltas[o]=1*error_o[o]#输出层反向传播（求导）

            for h in range(self.hidden_nodes): # 隐 藏 层
                for o in range(self.output_nodes):
#                     print('weight::',self.weights_hidden_to_output[h][o])
                    error_h[h]+=output_deltas[o]*self.weights_hidden_to_output[h][o]#计算隐藏层误差

#                 print('....')
#                 print(self.hidden_nodes_value[h])
#                 print(error_h[h])
                hidden_deltas[h]=self.delta_activation_function(self.hidden_nodes_value[h])*error_h[h]#隐藏层反向传播
#                 print(hidden_deltas[h])

            for h in range(self.hidden_nodes):
                for o in range(self.output_nodes):
                    self.change_to_fix_weights_h2o[h][o]+=output_deltas[o]*self.hidden_nodes_value[h]#累计隐藏层>>输出层的权重矩阵的调整量

            for i in range(self.input_nodes):
                for h in range(self.hidden_nodes):
#                     print("......")
#                     print(hidden_deltas[h])
#                     print(self.input_nodes_value[i])
#                     print(self.change_to_fix_weights_i2h[i][h])
                    self.change_to_fix_weights_i2h[i][h]+=hidden_deltas[h]*self.input_nodes_value[i]#累计输入层>>隐藏层的权重矩阵的调整量

            counter-=1
            if counter==0:#完成一个batch的输入和计算后，调整一次权重
                #调整隐藏层>>输出层权重
                for h in range(self.hidden_nodes):
                    for o in range(self.output_nodes):
                        self.weights_hidden_to_output[h][o] += self.learning_rate*self.change_to_fix_weights_h2o[h][o]

                #调整输入层>>隐藏层权重
                for i in range(self.input_nodes):
                    for h in range(self.hidden_nodes):
#                         print("......")
#                         print(self.weights_input_to_hidden[i][h])
#                         print(self.learning_rate)
#                         print(self.change_to_fix_weights_i2h[i][h])
                        self.weights_input_to_hidden[i][h] += self.learning_rate*self.change_to_fix_weights_i2h[i][h]
#                         print(self.weights_input_to_hidden[i][h])
                #将权值调整量归零，计数器复位，开始输入下一个batch
                self.change_to_fix_weights_h2o=[[0.0]*self.output_nodes]*self.hidden_nodes
                self.change_to_fix_weights_i2h=[[0.0]*self.hidden_nodes]*self.input_nodes
                counter=batch_size
        return self.weights_hidden_to_output

    def run(self, features):#完成一条数据的一次前向传递
        '''
            features: 1D array of feature values
        '''
#         print(self.input_nodes_value)
        for i in range(self.input_nodes):
            self.input_nodes_value[i]=features[i]
#             self.input_nodes_value[i]=self.activation_function(features[i])
#         print(self.input_nodes_value)

#         print(self.hidden_nodes_value)
        for h in range(self.hidden_nodes):
            temp=0
            for i in range(self.input_nodes):
                temp+=self.input_nodes_value[i]*self.weights_input_to_hidden[i][h]
            temp=self.activation_function(temp)
            self.hidden_nodes_value[h]=temp
#         print(self.hidden_nodes_value)

#         print(self.output_nodes_value)
        for o in range(self.output_nodes):
            temp=0
            for h in range(self.hidden_nodes):
                temp+=self.hidden_nodes_value[h]*self.weights_hidden_to_output[h][o]
#             temp=self.activation_function(temp)
            self.output_nodes_value[o]=temp
#         print(self.output_nodes_value)

        return self.output_nodes_value
        

单元测试：

import unittest

inputs = np.array([[0.5, -0.2, 0.1]])
targets = np.array([[0.4]])
test_w_i_h = np.array([[0.1, -0.2],
                       [0.4, 0.5],
                       [-0.3, 0.2]])
test_w_h_o = np.array([[0.3],
                       [-0.1]])

class TestMethods(unittest.TestCase):

    ##########
    # Unit tests for data loading
    ##########

    def test_data_path(self):
        # Test that file path to dataset has been unaltered
        self.assertTrue(data_path.lower() == 'bike-sharing-dataset/hour.csv')

    def test_data_loaded(self):
        # Test that data frame loaded
        self.assertTrue(isinstance(rides, pd.DataFrame))

    ##########
    # Unit tests for network functionality
    ##########

    def test_activation(self):
        network = NeuralNetwork(3, 2, 1, 0.5)
        # Test that the activation function is a sigmoid
        self.assertTrue(np.all(network.activation_function(0.5) == 1/(1+np.exp(-0.5))))

    def test_train(self):
        # Test that weights are updated correctly on training
        network = NeuralNetwork(3, 2, 1, 0.5)
        network.weights_input_to_hidden = test_w_i_h.copy()
        network.weights_hidden_to_output = test_w_h_o.copy()

        network.train(inputs, targets)
        print('@@@@test_train')
        print("$$$$$$$$1")
        print(network.weights_hidden_to_output)
        print(network.weights_input_to_hidden)

#         network.train(inputs,targets)

#         print("$$$$$$$$2")
#         print(network.weights_hidden_to_output)
#         print(network.weights_input_to_hidden)

        self.assertTrue(np.allclose(network.weights_hidden_to_output,
                                    np.array([[ 0.37275328],
                                              [-0.03172939]])))
        self.assertTrue(np.allclose(network.weights_input_to_hidden,
                                    np.array([[ 0.10562014, -0.20185996],
                                              [0.39775194, 0.50074398],
                                              [-0.29887597, 0.19962801]])))

    def test_run(self):
        # Test correctness of run method
        network = NeuralNetwork(3, 2, 1, 0.5)
        network.weights_input_to_hidden = test_w_i_h.copy()
        network.weights_hidden_to_output = test_w_h_o.copy()

        self.assertTrue(np.allclose(network.run(inputs[0]), 0.09998924))

suite = unittest.TestLoader().loadTestsFromModule(TestMethods())
unittest.TextTestRunner().run(suite)

结果：

结果虽然比较接近，但是代码比较丑陋，并没有用numpy的矩阵相乘，而是用for循环实现了矩阵乘法，代码复杂，而且都是串行的。