Error Back Propagation in BPNeuralNetwork及手写字体识别python版

　　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~

神经网络简介

　　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。

Error Back Propagation

优化的目标函数

　　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。

逐层求导过程

　　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)

　　=========================================================

小结

　　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)；

参考文献

[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 #