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   能量模型的概念从统计力学中得来,它描述着整个系统的某种状态,系统越有序,系统能量波动越小,趋*于*衡状态,系统越无序,能量波动越大。例如:一个孤立的物体,其内部各处的温度不尽相同,那么热就从温度较高的地方流向温度较低的地方,最后达到各处温度都相同的状态,也就是热*衡的状态。在统计力学中,系统处于某个状态的相对概率为,即玻尔兹曼因子,其中T表示温度,是玻尔兹曼常数,是状态的能量。玻尔兹曼因子本身并不是一个概率,因为它还没有归一化。为了把玻尔兹曼因子归一化,使其成为一个概率,我们把它除以系统所有可能的状态的玻尔兹曼因子之和Z,称为配分函数(partition function)。这便给出了玻尔兹曼分布。

    玻尔兹曼机(Boltzmann Machine,BM)是一种特殊形式的对数线性的马尔科夫随机场(Markov Random Field,MRF),即能量函数是自由变量的线性函数。 通过引入隐含单元,我们可以提升模型的表达能力,表示非常复杂的概率分布。限制性玻尔兹曼机(RBM)进一步加一些约束,在RBM中不存在可见单元与可见单元的链接,也不存在隐含单元与隐含单元的链接,如下图所示

    能量函数在限制玻尔兹曼机中定义为,b,c,W为模型的参数,b,c分别为可见层和隐含层的偏置,W为可见层与隐含层的链接权重

    有了上述三个公式我们可以使用最大似然估计来求解模型的参数:设 。把概率p(x)改写为

    由于可见单元V和不可见单元h条件独立,利用这一性质,我们可以得到:

    logistic回归估计v与h取一的概率:

    有了以上条件,我们可以推导出参数变化的梯度值:

    使用基于马尔可夫链的gibbs抽样,对于一个d维的随机向量x=(x1,x2,…xd),假设我们无法求得x的联合概率分布p(x),但我们知道给定x的其他分量是其第i个分量xi的条件分布,即p(xi|xi-),xi-=(x1,x2,…xi-1,xi+1…xd)。那么,我们可以从x的一个任意状态(如(x1(0),x2(0),…,xd(0)))开始,利用条件分布p(xi|xi-),迭代地对这状态的每个分量进行抽样,随着抽样次数n的增加,随机变量(x1(n),x2(n),…,xd(n))的概率分布将以n的几何级数的速度收敛到x的联合概率分布p(v)。

    基于RBM模型的对称结构,以及其中节点的条件独立行,我们可以使用Gibbs抽样方法得到服从RBM定义的分布的随机样本。在RBM中进行k步Gibbs抽样的具体算法为:用一个训练样本(或者可视节点的一个随机初始状态)初始化可视节点的状态v0,交替进行下面的抽样:

    理论上,参数的每次更新需要让上面的链条图形遍历一次,这样带来的性能损耗毫无疑问是不能承受的。

    Hinton教授提出一种改进方法叫做对比分歧(Contrastive Divergence),即CD-K。他指出CD没有必要等待链收敛,样本可以通过k步 的gibbs抽样完成,仅需要较少的抽样步数(实验中使用一步)就可以得到足够好的效果。

    下面给出RBM用到的CD-K算法伪代码。

     关于deeplearning的c++实现放到了github上,由于时间关系只是实现了大致框架,细节方面有待改善,也欢迎大家的参与https://github.com/loujiayu/deeplearning

  下面附上Geoff Hinton提供的关于RBM的matlab代码

% 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;

[numcases numdims numbatches]=size(batchdata);

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

% Initializing symmetric weights and biases. 
  vishid     = 0.1*randn(numdims, numhid);
  hidbiases  = zeros(1,numhid);
  visbiases  = zeros(1,numdims);

  poshidprobs = zeros(numcases,numhid);
  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);
end

for epoch = epoch:maxepoch,
 fprintf(1,'epoch %d\r',epoch); 
 errsum=0;
 for batch = 1:numbatches,
 fprintf(1,'epoch %d batch %d\r',epoch,batch); 

%%%%%%%%% START POSITIVE PHASE %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  data = batchdata(:,:,batch);
  poshidprobs = 1./(1 + exp(-data*vishid - repmat(hidbiases,numcases,1)));    
  batchposhidprobs(:,:,batch)=poshidprobs;
  posprods    = data' * poshidprobs;
  poshidact   = sum(poshidprobs);
  posvisact = sum(data);

%%%%%%%%% END OF POSITIVE PHASE  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  poshidstates = poshidprobs > rand(numcases,numhid);

%%%%%%%%% START NEGATIVE PHASE  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  negdata = 1./(1 + exp(-poshidstates*vishid' - repmat(visbiases,numcases,1)));
  neghidprobs = 1./(1 + exp(-negdata*vishid - repmat(hidbiases,numcases,1)));    
  negprods  = negdata'*neghidprobs;
  neghidact = sum(neghidprobs);
  negvisact = sum(negdata); 

%%%%%%%%% END OF NEGATIVE PHASE %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  err= sum(sum( (data-negdata).^2 ));
  errsum = err + errsum;

   if epoch>5,
     momentum=finalmomentum;
   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;

 

posted on 2014-02-28 13:51  Stinky tofu  阅读(3229)  评论(1编辑  收藏  举报