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DeepLearningToolBox学习——RBM(Restrict Boltzmann Machine) 1.ministdeepauto2. Rbm 4.makebatches5. backprop5. CG_MNIST6.minimize——共轭梯度下降的优化函数形式

下载地址：DeepLearningToolBox

     学习RBM代码之前，需要一些基本的RBM的知识。

     网上有很多参考资料，我借鉴一篇写的很好的系列文章，看下来也差不多能看懂了，博客地址：http://blog.csdn.net/itplus/article/details/19168937

 目录如下

（一）预备知识

（二）网络结构

（三）能量函数和概率分布

（四）对数似然函数

（五）梯度计算公式

（六）对比散度算法

（七）RBM 训练算法

（八）RBM 的评估

通过学习上面的系列文章，基本上理解了RBM的原理，接下来动手学习toolbox中对应的代码部分。本文参考诸多博客，感谢原文作者。

1.ministdeepauto

  这个code对应的原文是 Hition大牛science文章reducing the dimensionality of data with neural networks。  用MNIST数据库来进行深度的autoencoder压缩，用的是无监督学习，评价标准是重构误差值MSE。

% Version 1.000
%
% Code provided by Ruslan Salakhutdinov and Geoff Hinton  
%
% Permission is granted for anyone to copy, use, modify, or distribute this
% program and accompanying programs and documents for any purpose, provided
% this copyright notice is retained and prominently displayed, along with
% a note saying that the original programs are available from our 
% web page. 
% The programs and documents are distributed without any warranty, express or
% implied.  As the programs were written for research purposes only, they have
% not been tested to the degree that would be advisable in any important
% application.  All use of these programs is entirely at the user's own risk.


% This program pretrains a deep autoencoder for MNIST dataset
% You can set the maximum number of epochs for pretraining each layer
% and you can set the architecture of the multilayer net.

clc
clear all
close all

maxepoch=10; %In the Science paper we use maxepoch=50, but it works just fine. 
numhid=1000; numpen=500; numpen2=250; numopen=30;

fprintf(1,'Converting Raw files into Matlab format \n');
converter; % 转换数据为matlab的格式

fprintf(1,'Pretraining a deep autoencoder. \n');
fprintf(1,'The Science paper used 50 epochs. This uses %3i \n', maxepoch);

makebatches;
[numcases numdims numbatches]=size(batchdata);

fprintf(1,'Pretraining Layer 1 with RBM: %d-%d \n',numdims,numhid);
restart=1;
rbm;
hidrecbiases=hidbiases; %hidbiases为隐含层的偏置值
save mnistvh vishid hidrecbiases visbiases;
%保存每层的变量，分别为权值，隐含层偏置值，可视层偏置值

fprintf(1,'\nPretraining Layer 2 with RBM: %d-%d \n',numhid,numpen);
batchdata=batchposhidprobs; %batchposhidprobs为第一个rbm的输出概率值
numhid=numpen;
restart=1;
rbm;
hidpen=vishid; penrecbiases=hidbiases; hidgenbiases=visbiases;
save mnisthp hidpen penrecbiases hidgenbiases;
%mnisthp为所保存的文件名

fprintf(1,'\nPretraining Layer 3 with RBM: %d-%d \n',numpen,numpen2);
batchdata=batchposhidprobs;
numhid=numpen2;
restart=1;
rbm;
hidpen2=vishid; penrecbiases2=hidbiases; hidgenbiases2=visbiases;
save mnisthp2 hidpen2 penrecbiases2 hidgenbiases2;

fprintf(1,'\nPretraining Layer 4 with RBM: %d-%d \n',numpen2,numopen);
batchdata=batchposhidprobs;
numhid=numopen; 
restart=1;
rbmhidlinear;
hidtop=vishid; toprecbiases=hidbiases; topgenbiases=visbiases;
save mnistpo hidtop toprecbiases topgenbiases;

backprop; %Finetune

本次是训练4个隐含层的autoencoder深度网络结构，输入层维度为784维，4个隐含层维度分别为1000,500,250,30。整个网络权值的获得流程梳理如下：

  1. 首先训练第一个rbm网络，即输入层784维和第一个隐含层1000维构成的网络。采用的方法是rbm优化，这个过程用的是训练样本，优化完毕后，计算训练样本在隐含层的输出值。
  2. 利用1中的结果作为第2个rbm网络训练的输入值，同样用rbm网络来优化第2个rbm网络，并计算出网络的输出值。并且用同样的方法训练第3个rbm网络和第4个rbm网络。
  3. 将上面4个rbm网络展开连接成新的网络，且分成encoder和decoder部分。并用步骤1和2得到的网络值给这个新网络赋初值。
  4. 由于新网络中最后的输出和最初的输入节点数是相同的，所以可以将最初的输入值作为网络理论的输出标签值，然后采用BP算法计算网络的代价函数和代价函数的偏导数。
  5. 利用步骤3的初始值和步骤4的代价值和偏导值，采用共轭梯度下降法优化整个新网络，得到最终的网络权值。以上整个过程都是无监督的。

2. Rbm 

% Version 1.000 
%
% Code provided by Geoff Hinton and Ruslan Salakhutdinov 
%
% Permission is granted for anyone to copy, use, modify, or distribute this
% program and accompanying programs and documents for any purpose, provided
% this copyright notice is retained and prominently displayed, along with
% a note saying that the original programs are available from our
% web page.
% The programs and documents are distributed without any warranty, express or
% implied.  As the programs were written for research purposes only, they have
% not been tested to the degree that would be advisable in any important
% application.  All use of these programs is entirely at the user's own risk.

% This program trains Restricted Boltzmann Machine in which
% visible, binary, stochastic pixels are connected to
% hidden, binary, stochastic feature detectors using symmetrically
% weighted connections. Learning is done with 1-step Contrastive Divergence.   
% The program assumes that the following variables are set externally:
% maxepoch  -- maximum number of epochs
% numhid    -- number of hidden units 
% batchdata -- the data that is divided into batches (numcases numdims numbatches)
% restart   -- set to 1 if learning starts from beginning 

epsilonw      = 0.1;   % Learning rate for weights 
epsilonvb     = 0.1;   % Learning rate for biases of visible units 
epsilonhb     = 0.1;   % Learning rate for biases of hidden units 
weightcost  = 0.0002;   
initialmomentum  = 0.5;
finalmomentum    = 0.9;
 %由此可见这里隐含层和可视层的偏置值不是共用的，当然了，其权值是共用的
 
[numcases numdims numbatches]=size(batchdata);%[100,784,600]

if restart ==1,
  restart=0;
  epoch=1;

% Initializing symmetric weights and biases. 
  vishid     = 0.1*randn(numdims, numhid);%权值初始值随便给,784*1000
  hidbiases  = zeros(1,numhid);
  visbiases  = zeros(1,numdims);

  poshidprobs = zeros(numcases,numhid); %100*1000，单个batch正向传播时隐含层的输出概率
  neghidprobs = zeros(numcases,numhid);
  posprods    = zeros(numdims,numhid);
  negprods    = zeros(numdims,numhid);
  vishidinc  = zeros(numdims,numhid);
  hidbiasinc = zeros(1,numhid);
  visbiasinc = zeros(1,numdims);
  batchposhidprobs=zeros(numcases,numhid,numbatches);
  % 整个数据正向传播时隐含层的输出概率
end

for epoch = epoch:maxepoch,
 fprintf(1,'epoch %d\r',epoch); 
 errsum=0;
 for batch = 1:numbatches, %每次迭代都有遍历所有的batch
 fprintf(1,'epoch %d batch %d\r',epoch,batch); 

%%%%%%%%% START POSITIVE PHASE %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  data = batchdata(:,:,batch);
  % 每次迭代都需要取出一个batch的数据，每一行代表一个样本值（这里的数据是double的，不是01的，严格的说后面应将其01化）
  poshidprobs = 1./(1 + exp(-data*vishid - repmat(hidbiases,numcases,1))); 
  % 样本正向传播时隐含层节点的输出概率 
  batchposhidprobs(:,:,batch)=poshidprobs;
  posprods    = data' * poshidprobs;
  %784*1000，这个是求系统的能量值用的，矩阵中每个元素表示对应的可视层节点和隐含层节点的乘积（包含此次样本的数据对应值的累加）
  poshidact   = sum(poshidprobs);%针对样本值进行求和
  posvisact = sum(data);

%%%%%%%%% END OF POSITIVE PHASE  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  poshidstates = poshidprobs > rand(numcases,numhid);
  %将隐含层数据01化（此步骤在posprods之后进行），按照概率值大小来判定.

%%%%%%%%% START NEGATIVE PHASE  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  negdata = 1./(1 + exp(-poshidstates*vishid' - repmat(visbiases,numcases,1)));% 反向进行时的可视层数据
  neghidprobs = 1./(1 + exp(-negdata*vishid - repmat(hidbiases,numcases,1))); % 反向进行后又马上正向传播的隐含层概率值    
  negprods  = negdata'*neghidprobs;% 同理也是计算能量值用的，784*1000
  neghidact = sum(neghidprobs);
  negvisact = sum(negdata); 

%%%%%%%%% END OF NEGATIVE PHASE %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  err= sum(sum( (data-negdata).^2 ));% 重构后的差值
  errsum = err + errsum;

   if epoch>5,
     momentum=finalmomentum;
     %momentum为保持上一次权值更新增量的比例，如果迭代次数越少，则这个比例值可以稍微大一点
   else
     momentum=initialmomentum;
   end;

%%%%%%%%% UPDATE WEIGHTS AND BIASES %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
    vishidinc = momentum*vishidinc + ...
                epsilonw*( (posprods-negprods)/numcases - weightcost*vishid);
    visbiasinc = momentum*visbiasinc + (epsilonvb/numcases)*(posvisact-negvisact);
    hidbiasinc = momentum*hidbiasinc + (epsilonhb/numcases)*(poshidact-neghidact);

    vishid = vishid + vishidinc;
    visbiases = visbiases + visbiasinc;
    hidbiases = hidbiases + hidbiasinc;

%%%%%%%%%%%%%%%% END OF UPDATES %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

  end
  fprintf(1, 'epoch %4i error %6.1f  \n', epoch, errsum); 
end;

下面来看下在程序中大致实现RBM权值的优化步骤（假设是一个2层的RBM网络，即只有输入层和输出层，且这两层上的变量是二值变量）：

  1. 随机给网络初始化一个权值矩阵w和偏置向量b。
  2. 对可视层输入矩阵v正向传播，计算出隐含层的输出矩阵h，并计算出输入v和h对应节点乘积的均值矩阵
  3. 此时2中的输出h为概率值，将它随机01化为二值变量。
  4. 利用3中01化了的h方向传播计算出可视层的矩阵v’.(按照道理，这个v'应该是要01化的)
  5. 对v’进行正向传播计算出隐含层的矩阵h’，并计算出v’和h’对应节点乘积的均值矩阵。
  6. 用2中得到的均值矩阵减掉5中得到的均值矩阵，其结果作为对应权值增量的矩阵。
  7. 结合其对应的学习率，利用权值迭代公式对权值进行迭代。
  8. 重复计算2到7，直至收敛。

　　偏置值的优化步骤：

  1. 随机给网络初始化一个权值矩阵w和偏置向量b。
  2. 对可视层输入矩阵v正向传播，计算出隐含层的输出矩阵h，并计算v层样本的均值向量以及h层的均值向量。
  3. 此时2中的输出h为概率值，将它随机01化为二值变量。
  4. 利用3中01化了的h方向传播计算出可视层的矩阵v’.
  5. 对v’进行正向传播计算出隐含层的矩阵h’， 并计算v‘层样本的均值向量以及h’层的均值向量。
  6. 用2中得到的v方均值向量减掉5中得到的v’方的均值向量，其结果作为输入层v对应偏置的增值向量。用2中得到的h方均值向量减掉5中得到的h’方的均值向量，其结果作为输入层h对应偏置的增值向量。
  7. 结合其对应的学习率，利用权值迭代公式对偏置值进行迭代。
  8. 重复计算2到7，直至收敛。

　　当然了，权值更新和偏置值更新每次迭代都是同时进行的，所以应该是同时收敛的。并且在权值更新公式也可以稍微作下变形，比如加入momentum变量，即本次权值更新的增量会保留一部分上次更新权值的增量值。

3. converter

实现的功能是将样本集从.ubyte格式转换成.ascii格式，然后继续转换成.mat格式。

4.makebatches

实现的是将原本的2维数据集变成3维的，因为分了多个批次，另外1维表示的是批次。

5. backprop

反向传递误差

% Version 1.000
%
% Code provided by Ruslan Salakhutdinov and Geoff Hinton
%
% Permission is granted for anyone to copy, use, modify, or distribute this
% program and accompanying programs and documents for any purpose, provided
% this copyright notice is retained and prominently displayed, along with
% a note saying that the original programs are available from our
% web page.
% The programs and documents are distributed without any warranty, express or
% implied.  As the programs were written for research purposes only, they have
% not been tested to the degree that would be advisable in any important
% application.  All use of these programs is entirely at the user's own risk.

% This program fine-tunes an autoencoder with backpropagation.
% Weights of the autoencoder are going to be saved in mnist_weights.mat
% and trainig and test reconstruction errors in mnist_error.mat
% You can also set maxepoch, default value is 200 as in our paper.  

maxepoch=200;
fprintf(1,'\nFine-tuning deep autoencoder by minimizing cross entropy error. \n');%其微调通过最小化交叉熵来实现
fprintf(1,'60 batches of 1000 cases each. \n');

load mnistvh % 分别load4个rbm的参数
load mnisthp
load mnisthp2
load mnistpo 

makebatches;
[numcases numdims numbatches]=size(batchdata);
N=numcases; 

%%%% PREINITIALIZE WEIGHTS OF THE AUTOENCODER %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
w1=[vishid; hidrecbiases]; %分别装载每层的权值和偏置值，将它们作为一个整体
w2=[hidpen; penrecbiases];
w3=[hidpen2; penrecbiases2];
w4=[hidtop; toprecbiases];
w5=[hidtop'; topgenbiases]; 
w6=[hidpen2'; hidgenbiases2]; 
w7=[hidpen'; hidgenbiases]; 
w8=[vishid'; visbiases];

%%%%%%%%%% END OF PREINITIALIZATIO OF WEIGHTS  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

l1=size(w1,1)-1;%每个网络层中节点的个数
l2=size(w2,1)-1;
l3=size(w3,1)-1;
l4=size(w4,1)-1;
l5=size(w5,1)-1;
l6=size(w6,1)-1;
l7=size(w7,1)-1;
l8=size(w8,1)-1;
l9=l1;  %输出层节点和输入层的一样
test_err=[];
train_err=[];


for epoch = 1:maxepoch

%%%%%%%%%%%%%%%%%%%% COMPUTE TRAINING RECONSTRUCTION ERROR %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
err=0; 
[numcases numdims numbatches]=size(batchdata);
N=numcases;
 for batch = 1:numbatches
  data = [batchdata(:,:,batch)];
  data = [data ones(N,1)];  % b补上一维，因为有偏置项
  w1probs = 1./(1 + exp(-data*w1)); w1probs = [w1probs  ones(N,1)];;
  %正向传播，计算每一层的输出，且同时在输出上增加一维（值为常量1）
  w2probs = 1./(1 + exp(-w1probs*w2)); w2probs = [w2probs ones(N,1)];
  w3probs = 1./(1 + exp(-w2probs*w3)); w3probs = [w3probs  ones(N,1)];
  w4probs = w3probs*w4; w4probs = [w4probs  ones(N,1)];
  w5probs = 1./(1 + exp(-w4probs*w5)); w5probs = [w5probs  ones(N,1)];
  w6probs = 1./(1 + exp(-w5probs*w6)); w6probs = [w6probs  ones(N,1)];
  w7probs = 1./(1 + exp(-w6probs*w7)); w7probs = [w7probs  ones(N,1)];
  dataout = 1./(1 + exp(-w7probs*w8));
  err= err +  1/N*sum(sum( (data(:,1:end-1)-dataout).^2 )); %重构的误差值
  end
 train_err(epoch)=err/numbatches; %总的误差值（训练样本上）

%%%%%%%%%%%%%% END OF COMPUTING TRAINING RECONSTRUCTION ERROR %%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%% DISPLAY FIGURE TOP ROW REAL DATA BOTTOM ROW RECONSTRUCTIONS %%%%%%%%%%%%%%%%%%%%%%%%%
fprintf(1,'Displaying in figure 1: Top row - real data, Bottom row -- reconstructions \n');
output=[];
 for ii=1:15
  output = [output data(ii,1:end-1)' dataout(ii,:)']; %output为15（因为是显示15个数字）组，每组2列，分别为理论值和重构值
 end
   if epoch==1 
   close all 
   figure('Position',[100,600,1000,200]);
   else 
   figure(1)
   end 
   mnistdisp(output);
   drawnow;

%%%%%%%%%%%%%%%%%%%% COMPUTE TEST RECONSTRUCTION ERROR %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
[testnumcases testnumdims testnumbatches]=size(testbatchdata);
N=testnumcases;
err=0;
for batch = 1:testnumbatches
  data = [testbatchdata(:,:,batch)];
  data = [data ones(N,1)];
  w1probs = 1./(1 + exp(-data*w1)); w1probs = [w1probs  ones(N,1)];
  w2probs = 1./(1 + exp(-w1probs*w2)); w2probs = [w2probs ones(N,1)];
  w3probs = 1./(1 + exp(-w2probs*w3)); w3probs = [w3probs  ones(N,1)];
  w4probs = w3probs*w4; w4probs = [w4probs  ones(N,1)];
  w5probs = 1./(1 + exp(-w4probs*w5)); w5probs = [w5probs  ones(N,1)];
  w6probs = 1./(1 + exp(-w5probs*w6)); w6probs = [w6probs  ones(N,1)];
  w7probs = 1./(1 + exp(-w6probs*w7)); w7probs = [w7probs  ones(N,1)];
  dataout = 1./(1 + exp(-w7probs*w8));
  err = err +  1/N*sum(sum( (data(:,1:end-1)-dataout).^2 ));
  end
 test_err(epoch)=err/testnumbatches;
 fprintf(1,'Before epoch %d Train squared error: %6.3f Test squared error: %6.3f \t \t \n',epoch,train_err(epoch),test_err(epoch));

%%%%%%%%%%%%%% END OF COMPUTING TEST RECONSTRUCTION ERROR %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 tt=0;
 for batch = 1:numbatches/10 %测试样本numbatches是100
 fprintf(1,'epoch %d batch %d\r',epoch,batch);

%%%%%%%%%%% COMBINE 10 MINIBATCHES INTO 1 LARGER MINIBATCH %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 tt=tt+1; 
 data=[];
 for kk=1:10
  data=[data 
        batchdata(:,:,(tt-1)*10+kk)]; 
 end 

%%%%%%%%%%%%%%% PERFORM CONJUGATE GRADIENT WITH 3 LINESEARCHES %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%共轭梯度线性搜索
  max_iter=3;
  VV = [w1(:)' w2(:)' w3(:)' w4(:)' w5(:)' w6(:)' w7(:)' w8(:)']';;
  % 把所有权值（已经包括了偏置值）变成一个大的列向量
  Dim = [l1; l2; l3; l4; l5; l6; l7; l8; l9];
  %每层网络对应节点的个数（不包括偏置值）
  [X, fX] = minimize(VV,'CG_MNIST',max_iter,Dim,data);%该函数时使用共轭梯度的方法来对参数X进行优化
  
  w1 = reshape(X(1:(l1+1)*l2),l1+1,l2);
  xxx = (l1+1)*l2;
  w2 = reshape(X(xxx+1:xxx+(l2+1)*l3),l2+1,l3);
  xxx = xxx+(l2+1)*l3;
  w3 = reshape(X(xxx+1:xxx+(l3+1)*l4),l3+1,l4);
  xxx = xxx+(l3+1)*l4;
  w4 = reshape(X(xxx+1:xxx+(l4+1)*l5),l4+1,l5);
  xxx = xxx+(l4+1)*l5;
  w5 = reshape(X(xxx+1:xxx+(l5+1)*l6),l5+1,l6);
  xxx = xxx+(l5+1)*l6;
  w6 = reshape(X(xxx+1:xxx+(l6+1)*l7),l6+1,l7);
  xxx = xxx+(l6+1)*l7;
  w7 = reshape(X(xxx+1:xxx+(l7+1)*l8),l7+1,l8);
  xxx = xxx+(l7+1)*l8;
  w8 = reshape(X(xxx+1:xxx+(l8+1)*l9),l8+1,l9);
%依次重新赋值为优化后的参数
%%%%%%%%%%%%%%% END OF CONJUGATE GRADIENT WITH 3 LINESEARCHES %%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 end

 save mnist_weights w1 w2 w3 w4 w5 w6 w7 w8 %前面一个是文件名
 save mnist_error test_err train_err;

end

5. CG_MNIST

 函数CG_MNIST形式如下：

　　function [f, df] = CG_MNIST(VV,Dim,XX);

　　该函数实现的功能是计算网络代价函数值f，以及f对网络中各个参数值的偏导数df，权值和偏置值是同时处理。其中参数VV为网络中所有参数构成的列向量，参数Dim为每层网络的节点数构成的向量，XX为训练样本集合。f和df分别表示网络的代价函数和偏导函数值。 

6.minimize——共轭梯度下降的优化函数形式

　　[X, fX, i] = minimize(X, f, length, P1, P2, P3, ... )

　　该函数时使用共轭梯度的方法来对参数X进行优化，所以X是网络的参数值，为一个列向量。f是一个函数的名称，它主要是用来计算网络中的代价函数以及代价函数对各个参数X的偏导函数，f的参数值分别为X，以及minimize函数后面的P1,P2,P3,…使用共轭梯度法进行优化的最大线性搜索长度为length。返回值X为找到的最优参数，fX为在此最优参数X下的代价函数，i为线性搜索的长度（即迭代的次数）。

本文参考如下博客：

http://www.cnblogs.com/tornadomeet/archive/2013/04/30/3052349.html

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