受限玻尔兹曼机(Restricted Boltzmann Machine)

作者：凯鲁嘎吉 - 博客园 http://www.cnblogs.com/kailugaji/

 1. 生成模型

 2. 参数学习

 3. 对比散度学习算法

    由于受限玻尔兹曼机的特殊结构,因此可以使用一种比吉布斯采样更有效 的学习算法,即对比散度(Contrastive Divergence)对比散度算法仅需k步吉布斯采样。为了提高效率,对比散度算法用一个训练样本作为可观测向量的初始值。然后,交替对可观测向量和隐藏向量进行吉布斯采样,不需要等到收敛,只需要k步就足够了。这就是CD-k 算法。通常,k = 1就可以学得很好。对比散度的流程如算法12.1所示。

 4. MATLAB程序解读

% 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  -- 最大迭代次数maximum number of epochs
% numhid    -- 隐含层神经元数number of hidden units 
% batchdata -- 分批后的训练数据集the data that is divided into batches (numcases numdims numbatches)
% restart   -- 如果从第1层开始学习，就置restart为1set to 1 if learning starts from beginning 

%作用：训练RBM，利用1步CD算法 直接调用权值迭代公式不使用反向传播 
%可见的、二元的、随机的像素通过对称加权连接连接到隐藏的、二元的、随机的特征检测器
epsilonw      = 0.1;   % Learning rate for weights 权重学习率 alpha
epsilonvb     = 0.1;   % Learning rate for biases of visible units 可视层偏置学习率 alpha
epsilonhb     = 0.1;   % Learning rate for biases of hidden units 隐藏层偏置学习率 alpha
weightcost  = 0.0002;   %权衰减，用于防止出现过拟合
initialmomentum  = 0.5; %动量项学习率，用于克服收敛速度和算法的不稳定性之间的矛盾
finalmomentum    = 0.9;

[numcases numdims numbatches]=size(batchdata);%[numcases numdims numbatches]＝[每批中的样本数 每个样本的维数 训练样本批数]

if restart ==1  %是否为重新开始即从头训练
  restart=0;
  epoch=1;

% Initializing symmetric weights and biases. 初始化权重和两层偏置
  vishid     = 0.1*randn(numdims, numhid);% 连接权值Wij 784*1000
  hidbiases  = zeros(1,numhid);% 隐含层偏置项bi
  visbiases  = zeros(1,numdims);% 可视化层偏置项aj

  poshidprobs = zeros(numcases,numhid); %样本数*隐藏层NN数，隐藏层输出p(h1|v0)对应每个样本有一个输出 100*1000
  neghidprobs = zeros(numcases,numhid);  %重构数据驱动的隐藏层
  posprods    = zeros(numdims,numhid);  % 表示p(h1|v0)*v0，用于更新Wij即<vihj>data 784*1000
  negprods    = zeros(numdims,numhid);  %<vihj>recon 
  vishidinc  = zeros(numdims,numhid);  % 权值更新的增量 ΔW
  hidbiasinc = zeros(1,numhid);  % 隐含层偏置项更新的增量 1*1000 Δb
  visbiasinc = zeros(1,numdims);  % 可视化层偏置项更新的增量 1*784 Δa
  batchposhidprobs=zeros(numcases,numhid,numbatches);  % 整个数据隐含层的输出  每批样本数*隐含层维度*批数
end

for epoch = epoch:maxepoch  %每个迭代周期
 fprintf(1,'epoch %d\r',epoch); 
 errsum=0;
 for batch = 1:numbatches  %每一批样本
 fprintf(1,'epoch %d batch %d\r',epoch,batch); 
%%CD-1
%%%%%%%%% START POSITIVE PHASE 正向梯度%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  data = batchdata(:,:,batch);  %data里是100个图片数据
  poshidprobs = 1./(1 + exp(-data*vishid - repmat(hidbiases,numcases,1)));  %隐藏层输出p(h=1|v0)=sigmod函数=1/(1+exp(-wx-b)) 根据这个分布采集一个隐变量h
  batchposhidprobs(:,:,batch)=poshidprobs;  %将输出存入一个三位数组
  posprods    = data' * poshidprobs;  %p(h|v0)*v0 更新权重时会使用到 计算正向梯度vh'
  poshidact   = sum(poshidprobs);  %隐藏层中神经元概率和，在更新隐藏层偏置时会使用到
  posvisact = sum(data);  %可视层中神经元概率和，在更新可视层偏置时会使用到
%%%%%%%%% END OF POSITIVE PHASE  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%gibbs采样
  poshidstates = poshidprobs > rand(numcases,numhid);  %将隐藏层输出01化表示，大于随机概率的置1，小于随机概率的置0，gibbs抽样,设定状态 

%%%%%%%%% START NEGATIVE PHASE 反向梯度%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  negdata = 1./(1 + exp(-poshidstates*vishid' - repmat(visbiases,numcases,1)));  %01化表示之后算vt=p(vt|ht-1)重构的数据 p(v=1|h)=sigmod(W*h+a)  采集重构的可见变量v'
  neghidprobs = 1./(1 + exp(-negdata*vishid - repmat(hidbiases,numcases,1)));    %ht=p(h|vt)使用重构数据隐藏层的输出 p(h=1|v)=sigmod(W'*v+b) 采样一个h'  
  negprods  = negdata'*neghidprobs;  %计算反向梯度v'h';
  neghidact = sum(neghidprobs);
  negvisact = sum(negdata); 
%%%%%%%%% END OF NEGATIVE PHASE %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%更新参数
  err= sum(sum( (data-negdata).^2 ));  %整批数据的误差 ||v-v'||^2
  errsum = err + errsum;

   if epoch>5  %迭代次数不同调整冲量
     momentum=finalmomentum;
   else
     momentum=initialmomentum;
   end

%%%%%%%%% UPDATE WEIGHTS AND BIASES 更新权重和偏置%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
    vishidinc = momentum*vishidinc + ...
                epsilonw*( (posprods-negprods)/numcases - weightcost*vishid); %权重的增量 ΔW=alpha*(vh'-v'h')
    visbiasinc = momentum*visbiasinc + (epsilonvb/numcases)*(posvisact-negvisact);  %可视层增量 Δa=alpha*(v-v')
    hidbiasinc = momentum*hidbiasinc + (epsilonhb/numcases)*(poshidact-neghidact);  %隐含层增量 Δb=alpha*(h-h')

    vishid = vishid + vishidinc; %a=a+Δa
    visbiases = visbiases + visbiasinc; %W=W+ΔW
    hidbiases = hidbiases + hidbiasinc;  %b=b+Δb
%%%%%%%%%%%%%%%% END OF UPDATES %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

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

5. 参考文献

[1] 邱锡鹏: 神经网络与深度学习

[2] Salakhutdinov R, Hinton G. Deep boltzmann machines[C]//Artificial intelligence and statistics. 2009: 448-455.

[3] Hinton, Training a deep autoencoder or a classifier on MNIST digits. 2006.