神经网络这么牛掰，归功于其强大的学习能力，而神经网络模型的好坏则依赖于Error Back Propagation 算法的强力支撑，现在对其进行一下推导，知其然知其所以然~
神经网络，顾名思义是尝试去模仿动物神经系统的一种网络模型。举个例子，如下图：
Zj 表示单元 j 的输入，hj表示单元 j 的输出，h(Zj)表示激活函数。 具体的神经网络的一些优势劣势之类的就不在这里细说了，大家可以自己去看，现在重点讲一下，神经网络里面的参数调整与误差的关系， 为什么叫误差后传，以及Error Back Progapation的推导核心。BPNN模型 f(x;w) 的好坏是用误差的期望 E[(f(x;w)−y)2] 来评估的。 这样模型最优参数的计算，就成了 woptimal=minw{E[(f(x;w)−y)2]} ，于是抽象为数学中的最值问题。 之前也有讲过，参数寻优的方法中，有借助于梯度来不断迭代找到最优参数的。详细见：http://blog.csdn.net/yujianmin1990/article/details/47461287 现在，也想对神经网络模型函数的参数 w 求导，以期望能够不断迭代w，从而得到整个模型的最优参数解。 不同于一般模型的是，神经网络是多层参数递进叠加到一起的，后面一层输入量与前面一层的输出量紧密联系。因此，妄图直接对所有层间的权重 WL 同时求偏导不是明智的做法. 如果前后两层间的权重偏导数存在着一点关系，岂不是很美妙，BP的设想正是基于此~ 神经网络的最小化目标函数是：
Em=1m∑i=1mError(xi) 其中Error(xi)=12∑k(dk(xi)−yk(xi))2 是单样本误差期望的衡量， k 表示输出单元个数。 dk(xi)表示在预测 xi 时，第k个输出单元上的预测值（输出值）， yk(xi) 表示真实样本，在第k个输出单元的真实值。1）先计算一下单个样本误差对上面图中第三层权重 Wkj 的偏导： ∂E∂W(L)kj=∂∂W(L)kj[12∑k(dk−yk)2]=(dk−yk)∗∂dk∂W(L)kj 已知： d(L)k(zk)=h(L)k(zk)=h(∑jWkj∗h(L−1)j+b(L)k) =(dk−yk)∗∂h(L)k(zk)∂W(L)kj=(dk−yk)∗∂h(L)k(zk)∂zk∗∂zkW(L)kj =(dk−yk)∗h′(L)k(zk)∗h(L−1)j 2）再计算一下误差期望对上面图中第二层权重 Wji 的偏导： Error缩写为E ∂E∂W(L−1)ji=∂∂W(L−1)ji[12∑k(dk−yk)2] =∑k(dk−yk)∗∂∂W(L−1)ji[h(L)k(zk)]=∑k(dk−yk)∗∂h(L)k(zk)∂zk∗∂zk∂W(L−1)ji zk=∑jW(L)kjh(L−1)j(zj)+b(L)k ;于是 ∂zk∂W(L−1)ji=W(L)kj⋅∂h(L−1)j(zj)∂zj⋅∂zj∂W(L−1)ji zj=∑iW(L−1)jihL−2i(zi)+b(L−1)j ;于是 ∂zj∂W(L−1)ji=hL−2i(zi) 得到最终： ∂E∂W(L−1)ji=∑k(dk−yk)∗∂h(L)(zk)∂zk∗[W(L)kj⋅∂h(L−1)(zj)∂zj⋅h(L−2)i(zi)] 3）我们将两层参数的偏导放到一块比较异同 ⎧⎩⎨⎪⎪⎪⎪∂E∂W(L)kj=(dk−yk)∗h′(L)k∗h(L−1)j ∂E∂W(L−1)ji=∑k(dk−yk)∗h′(L)k∗[W(L)kj⋅h′(L−1)j⋅h(L−2)i] 假设： ⎧⎩⎨δ(L)k=(dk−yk)∗h′(L)kδ(L−1)j=∑k[δ(L)kWkj]⋅h′(L−1)j将δ通过Wkj向输入的方向传递 再放到一起比较一下，则两者更像了：
⎧⎩⎨⎪⎪⎪⎪∂E∂W(L)kj=δ(L)k∗h(L−1)j∂E∂W(L−1)ji=δ(L−1)j∗h(L−2)i其中,δ(L)k=(dk−yk)∗h′(L)k其中,δ(L−1)j=∑k[δ(L)kWkj]⋅h′(L−1)j δ(L)k 与 δ(L−1)j 的关系，可以明确地看到，是将后面的 δ(L) 经过神经网络直接往输入方向计算得到之前的 δ(L−1) 的。 对偏移 b 求导数，得到∂E∂b(L)k=δ(L)k； ∂E∂b(L−1)j=δ(L−1)j 继续往下推， δ(L−2)n=∑j[δ(L−1)jWjn]⋅h′(L−2)n ，可以推广到更多层。 这样我们看到在最后一层的误差 d−y 通过层间权重和 h′ 逐层往前传递，这也是 误差后传的出处。 4）推导规律总结：notice ========================================================= W(L) 记做第 L 层的权重，连接L和 L+1 之间的权重， b(L) 作为本层计算偏移。 a）计算 L+1 层的输出误差： a.1） δ′(L+1)={∑W(L+2)δ(L+2)h−y,后层传递过来的δ,本层直接差h−y(最后一层时) a.2） δ(L+1)=δ′(L+1)∗h′(L+1) 乘以本层的激活函数的导数。 b）更新权重： ΔW(L)=h(L)∗δ(L+1) 本层输出*下层的 δ 。 c）更新偏 ： Δb(L)=δ(L+1) =========================================================1) BPNN主要是想通过参数求偏导数，然后迭代的方法寻找最优参数解。 逐层求偏导，结果发现层间的权重偏导是存在逐层递进关系的。 2) 另，一个隐层的BPNN最好调，两个隐层的BPNN稍微难调点，三层就开始难了。 代码仅附单隐层和双隐层的BPNN的参数，BP类是支持n层的。 附：Error Back Propagation 对梯度加上动量优化后的伪码。 对所有的层2≤l≤L，设ΔW(l)，这里ΔW(l)为全零矩阵； For i=1:m， 使用反向传播算法，计算各层神经元权值的梯度矩阵?W(l)(i)； 计算ΔW(l)=ΔW(l)+?W(l)(i)； (在Rumelhart的论文[1]中建议对ΔW(l)如下处理，以加快下降速度) (ΔW(l):=−εΔW(l)currentIter+αΔW(l)previousIter) 更新权值和偏置： 计算W(l)=W(l)+1mΔW(l)；
[1] Learning Representations by Back-propagating Errors 这个文章里并没有给出误差后传的样子，是不是我看得不细致啊。 [2] A Gentle Introduction to Backpropagation [3] http://www.cnblogs.com/biaoyu/archive/2015/06/20/4591304.html
BPNN的实现对手写字体的识别，python代码实现(96%准确率)：https://github.com/jxyyjm/test_nn/blob/master/src/mine_bpnn_epoch.py sklearn的NN代码（不是BPNN），对手写字体识别(98%准确率)：https://github.com/jxyyjm/test_nn/blob/master/src/sklearn_nn.py
#!/usr/bin/python # -*- coding:utf-8 -* ## @time : 2017-06-17 ## @author : yujianmin ## @reference : http://blog.csdn.net/yujianmin1990/article/details/49935007 ## @what-to-do : try to make a any-layer-nn by hand (one-input-layer; any-hidden-layer; one-output-layer) from __future__ import division from __future__ import print_function import logging import numpy as np from sklearn import metrics from tensorflow.examples.tutorials.mnist import input_data ## =========================================== ## ''' == input-layer == | == hidden-layer-1 == | == hidden-layer-2 == | == output-layer == I[0]-W[0]B[0]-P[0] | I[1]-W[1]B[1]-P[1] | I[2]-W[2]B[2]-P[2] | I[3]-W[3]B[3]-P[3] E[0]-DH[0]-D[0] | E[1]-DH[1]-D[1] | E[2]-DH[2]-D[2] | E[3]-DH[3]-D[3] DW[0]-DB[0] | DW[1]-DB[1] | DW[2]-DB[2] | DW[3]-DB[3] ''' # I--input; W--weight; P--probabilty # E--error; DH-delt_active_function; D--delt # DW--delt_W; DB--delt_B ## =========================================== ## class CMyBPNN: def __init__(self, hidden_nodes_list=[10, 10], batch_size=100, epoch=100, lr=0.5): self.train_data = '' self.test_data = '' self.model = '' self.W = [] self.B = [] self.C = [] self.middle_res = {} self.lr = lr self.epoch = epoch self.batch_size = batch_size self.layer_num = len(hidden_nodes_list)+2 self.hidden_nodes_list = hidden_nodes_list self.middle_res_file = './middle.res1' def __del__(self): self.train_data = '' self.test_data = '' self.model = '' self.W = [] self.B = [] self.C = [] self.middle_res = {} self.lr = '' self.epoch = '' self.batch_size = '' self.layer_num = '' self.hidden_nodes_list = [] self.middle_res_file = '' def read_data(self): mnist = input_data.read_data_sets('./MNIST_data', one_hot=True) self.train_data = mnist.train self.test_data = mnist.test def sigmoid(self, x): return 1/(1+np.exp(-x)) def delt_sigmoid(self, x): #return -np.exp(-x)/((1+np.exp(-x))**2) return self.sigmoid(x) * (1-self.sigmoid(x)) def get_label_pred(self, pred_mat): return np.argmax(pred_mat, axis=1) def compute_layer_input(self, layer_num): input = self.middle_res['layer_prob'][layer_num-1] return np.dot(input, self.W[layer_num-1]) + self.B[layer_num-1] def compute_layer_active(self, layer_num): return self.sigmoid(self.middle_res['layer_input'][layer_num]) def compute_layer_delt(self, layer_num): return self.delt_sigmoid(self.middle_res['layer_input'][layer_num]) def compute_diff(self, A, B): if len(A) != len(B): print ('A.shape:', A.shape, 'B.shape:', B.shape) return False error_mean= np.mean((A-B)**2) if error_mean>0: print ('(A-B)**2 = ', error_mean) return False else: return True def initial_weight_parameters(self): input_node_num = self.train_data.images.shape[1] output_node_num = self.train_data.labels.shape[1] ## 构造各层W (输入, 输出) node-num-pair 的 list ## node_num_pair = [(input_node_num, self.hidden_nodes_list[0])] for i in xrange(len(self.hidden_nodes_list)): if i == len(self.hidden_nodes_list)-1: node_num_pair.append((self.hidden_nodes_list[i], output_node_num)) else: node_num_pair.append((self.hidden_nodes_list[i], self.hidden_nodes_list[i+1])) for node_num_i, node_num_o in node_num_pair: W = np.reshape(np.array(np.random.normal(0, 0.001, node_num_i*node_num_o)), (node_num_i, node_num_o)) B = np.array(np.random.normal(0, 0.001, node_num_o)) ## 正向时的 Bias ## self.W.append(W) self.B.append(B) ## W,B,C len is self.layer_num-1 ## def initial_middle_parameters(self): self.middle_res['delt_W'] = [np.zeros_like(i) for i in self.W] ## 层与层之间的连接权重 # self.middle_res['delt_B'] = [np.zeros_like(i) for i in self.B] ## 前向传播时的偏倚量 # self.middle_res['layer_input'] = ['' for i in xrange(self.layer_num) ] ## 前向传播时的每层输入 # self.middle_res['layer_prob'] = ['' for i in xrange(self.layer_num) ] ## 前向传播时的每层激活概率 # self.middle_res['layer_delt'] = ['' for i in xrange(self.layer_num) ] ## 每层误差回传时，的起始误差 # self.middle_res['layer_wucha'] = ['' for i in xrange(self.layer_num) ] ## 每层由后层传递过来的误差 # self.middle_res['layer_active_delt'] = ['' for i in xrange(self.layer_num) ] ## 前向传播的激活后导数 # def forward_propagate(self, batch_x): ## 前向依层，计算 --> output ## 并保存中间结果 ## ## 0-layer ====== 1-layer; ====== 2-layer; ===== output-layer ## ## I[0]-W[0]-B[0] I[1]-W[1]-B[1] I[2]-W[2]-B[2] for layer_num in xrange(self.layer_num): if layer_num == 0: self.middle_res['layer_input'][layer_num] = batch_x self.middle_res['layer_prob'][layer_num] = batch_x self.middle_res['layer_active_delt'][layer_num] = batch_x else: self.middle_res['layer_input'][layer_num] = self.compute_layer_input(layer_num) self.middle_res['layer_prob'][layer_num] = self.compute_layer_active(layer_num) self.middle_res['layer_active_delt'][layer_num]= self.compute_layer_delt(layer_num) def compute_output_prob(self, x): for layer_num in xrange(self.layer_num): if layer_num == 0: output = x else: layer_num = layer_num -1 output = self.sigmoid(np.dot(output, self.W[layer_num])+self.B[layer_num]) return output def backward_propagate(self, batch_y): ## 后向依层，计算 --> delt ## 并保存中间结果 ## for layer_num in xrange(self.layer_num, 0, -1): layer_num = layer_num - 1 if layer_num == (self.layer_num -1): self.middle_res['layer_wucha'][layer_num] = self.middle_res['layer_prob'][layer_num] - batch_y self.middle_res['layer_delt'][layer_num] = \ self.middle_res['layer_wucha'][layer_num] * self.middle_res['layer_active_delt'][layer_num] else: self.middle_res['layer_wucha'][layer_num] = \ np.dot(self.middle_res['layer_delt'][layer_num+1], self.W[layer_num].T) self.middle_res['layer_delt'][layer_num] = \ self.middle_res['layer_wucha'][layer_num] * self.middle_res['layer_active_delt'][layer_num] self.middle_res['delt_W'][layer_num] = \ np.dot(self.middle_res['layer_prob'][layer_num].T, self.middle_res['layer_delt'][layer_num+1])/self.batch_size self.middle_res['delt_B'][layer_num] = np.mean(self.middle_res['layer_delt'][layer_num+1], axis=0) def update_weight(self): # for convient, compute from input to output dir # for layer_num in xrange(self.layer_num-1): self.W[layer_num] -= self.lr * self.middle_res['delt_W'][layer_num] self.B[layer_num] -= self.lr * self.middle_res['delt_B'][layer_num] def my_bpnn(self): self.read_data() self.initial_weight_parameters() self.initial_middle_parameters() iter_num= self.epoch*int(self.train_data.images.shape[0]/self.batch_size) for i in xrange(iter_num): batch_x, batch_y = self.train_data.next_batch(self.batch_size) # 1) compute predict-y self.forward_propagate(batch_x) # 2) compute delta-each-layer self.backward_propagate(batch_y) # 3) update the par-w-B self.update_weight() # 4) training is doing... if i%100 == 0: batch_x_prob = self.middle_res['layer_prob'][self.layer_num-1] test_x_prob = self.compute_output_prob(self.test_data.images) batch_acc, batch_confMat = self.compute_accuracy_confusionMat(batch_y, batch_x_prob) test_acc, test_confMat = self.compute_accuracy_confusionMat(self.test_data.labels, test_x_prob) print ('epoch:', round(i/self.train_data.images.shape[0], 1), 'iter:', i, 'train_batch_accuracy:', batch_acc, 'test_accuracy', test_acc) def save_middle_res(self, string_head): handle_w = open(self.middle_res_file, 'aw') for k, v in self.middle_res.iteritems(): handle_w.write(str(string_head)+'\n') handle_w.write(str(k)+'\n') if isinstance(v, list): num = 0 for i in v: try: shape = i.shape except: shape = '0' handle_w.write('v['+str(num)+'],'+str(shape)+'\n') handle_w.write(str(i)+'\n') num +=1 num = 0 for i in self.W: try: shape = i.shape except: shape = '0' handle_w.write('W['+str(num)+'],'+str(shape)+'\n') handle_w.write(str(i)+'\n') num +=1 num = 0 for i in self.B: try: shape = i.shape except: shape = '0' handle_w.write('B['+str(num)+'],'+str(shape)+'\n') handle_w.write(str(i)+'\n') num +=1 handle_w.close() def print_para_shape(self): for k, v in self.middle_res.iteritems(): str_save = '' if isinstance(v, list): num = 0 for i in v: try: shape = i.shape except: shape = '0' str_save += str(k) +' v['+str(num)+'],'+str(shape)+'\t|\t' num +=1 else: str_save = str(k)+':'+str(v) print (str_save.strip('|\t')) num = 0; str_save = '' for i in self.W: try: shape = i.shape except: shape = '0' str_save += 'W['+str(num)+'],'+str(shape)+'\t|\t' num +=1 print (str_save.strip('|\t')) num = 0; str_save = '' for i in self.B: try: shape = i.shape except: shape = '0' str_save += 'B['+str(num)+'],'+str(shape)+'\t|\t' num +=1 print (str_save.strip('|\t')) def compute_accuracy_confusionMat(self, real_mat, pred_mat): pred_label= self.get_label_pred(pred_mat) real_label= self.get_label_pred(real_mat) accuracy = metrics.accuracy_score(real_label, pred_label) confu_mat = metrics.confusion_matrix(real_label, pred_label, np.unique(real_label)) return accuracy, confu_mat if __name__=='__main__': CTest = CMyBPNN(hidden_nodes_list=[128], batch_size=100, epoch=100, lr=0.5) CTest.my_bpnn() # epoch: 100 iter: 549800 train_batch_accuracy: 0.98 test_accuracy 0.9717 # [100], batch_size=100, lr=0.05 # # epoch: 3 iter: 201500 train_batch_accuracy: 1.0 test_accuracy 0.9788 # [100], batch_size=100, lr=0.5 # #CTest = CMyBPNN(hidden_nodes_list=[250, 100], batch_size=150, epoch=50, lr=2.0) #CTest.my_bpnn() ## 两层以上hidden-layer就不是特别好调了 ## # epoch: 50 iter: 54900 train_batch_accuracy: 1.0 test_accuracy 0.9796 # [250, 100], batch_size=50, lr==2.0 #