受限玻爾茲曼機(Restricted Boltzmann Machine)
作者：凱魯嘎吉 - 博客園 http://www.cnblogs.com/kailugaji/
1. 生成模型
2. 參數學習
3. 對比散度學習算法
由於受限玻爾茲曼機的特殊結構,因此可以使用一種比吉布斯采樣更有效 的學習算法,即對比散度(Contrastive Divergence)對比散度算法僅需k步吉布斯采樣。為了提高效率,對比散度算法用一個訓練樣本作為可觀測向量的初始值。然后,交替對可觀測向量和隱藏向量進行吉布斯采樣,不需要等到收斂,只需要k步就足夠了。這就是CD-k 算法。通常,k = 1就可以學得很好。對比散度的流程如算法12.1所示。
4. MATLAB程序解讀
% maxepoch -- 最大迭代次數maximum number of epochs
% numhid -- 隱含層神經元數number of hidden units
% batchdata -- 分批后的訓練數據集the data that is divided into batches (numcases numdims numbatches)
% restart -- 如果從第1層開始學習，就置restart為1set to 1 if learning starts from beginning
%作用：訓練RBM，利用1步CD算法 直接調用權值迭代公式不使用反向傳播
%可見的、二元的、隨機的像素通過對稱加權連接連接到隱藏的、二元的、隨機的特征檢測器
epsilonw = 0.1; % Learning rate for weights 權重學習率 alpha
epsilonvb = 0.1; % Learning rate for biases of visible units 可視層偏置學習率 alpha
epsilonhb = 0.1; % Learning rate for biases of hidden units 隱藏層偏置學習率 alpha
weightcost = 0.0002; %權衰減，用於防止出現過擬合
initialmomentum = 0.5; %動量項學習率，用於克服收斂速度和算法的不穩定性之間的矛盾
finalmomentum = 0.9;
[numcases numdims numbatches]=size(batchdata);%[numcases numdims numbatches]＝[每批中的樣本數 每個樣本的維數 訓練樣本批數]
if restart ==1 %是否為重新開始即從頭訓練
restart=0;
epoch=1;
% Initializing symmetric weights and biases. 初始化權重和兩層偏置
vishid = 0.1*randn(numdims, numhid);% 連接權值Wij 784*1000
hidbiases = zeros(1,numhid);% 隱含層偏置項bi
visbiases = zeros(1,numdims);% 可視化層偏置項aj
poshidprobs = zeros(numcases,numhid); %樣本數*隱藏層NN數，隱藏層輸出p(h1|v0)對應每個樣本有一個輸出 100*1000
neghidprobs = zeros(numcases,numhid); %重構數據驅動的隱藏層
posprods = zeros(numdims,numhid); % 表示p(h1|v0)*v0，用於更新Wij即<vihj>data 784*1000
negprods = zeros(numdims,numhid); %<vihj>recon
vishidinc = zeros(numdims,numhid); % 權值更新的增量 ΔW
hidbiasinc = zeros(1,numhid); % 隱含層偏置項更新的增量 1*1000 Δb
visbiasinc = zeros(1,numdims); % 可視化層偏置項更新的增量 1*784 Δa
batchposhidprobs=zeros(numcases,numhid,numbatches); % 整個數據隱含層的輸出 每批樣本數*隱含層維度*批數
end
for epoch = epoch:maxepoch %每個迭代周期
fprintf(1,'epoch %d\r',epoch);
errsum=0;
for batch = 1:numbatches %每一批樣本
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個圖片數據
poshidprobs = 1./(1 + exp(-data*vishid - repmat(hidbiases,numcases,1))); %隱藏層輸出p(h=1|v0)=sigmod函數=1/(1+exp(-wx-b)) 根據這個分布采集一個隱變量h
batchposhidprobs(:,:,batch)=poshidprobs; %將輸出存入一個三位數組
posprods = data' * poshidprobs; %p(h|v0)*v0 更新權重時會使用到 計算正向梯度vh'
poshidact = sum(poshidprobs); %隱藏層中神經元概率和，在更新隱藏層偏置時會使用到
posvisact = sum(data); %可視層中神經元概率和，在更新可視層偏置時會使用到
%%%%%%%%% END OF POSITIVE PHASE %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%gibbs采樣
poshidstates = poshidprobs > rand(numcases,numhid); %將隱藏層輸出01化表示，大於隨機概率的置1，小於隨機概率的置0，gibbs抽樣,設定狀態
%%%%%%%%% 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)重構的數據 p(v=1|h)=sigmod(W*h+a) 采集重構的可見變量v'
neghidprobs = 1./(1 + exp(-negdata*vishid - repmat(hidbiases,numcases,1))); %ht=p(h|vt)使用重構數據隱藏層的輸出 p(h=1|v)=sigmod(W'*v+b) 采樣一個h'
negprods = negdata'*neghidprobs; %計算反向梯度v'h';
neghidact = sum(neghidprobs);
negvisact = sum(negdata);
%%%%%%%%% END OF NEGATIVE PHASE %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%更新參數
err= sum(sum( (data-negdata).^2 )); %整批數據的誤差 ||v-v'||^2
errsum = err + errsum;
if epoch>5 %迭代次數不同調整沖量
momentum=finalmomentum;
else
momentum=initialmomentum;
end
%%%%%%%%% UPDATE WEIGHTS AND BIASES 更新權重和偏置%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
vishidinc = momentum*vishidinc + ...
epsilonw*( (posprods-negprods)/numcases - weightcost*vishid); %權重的增量 ΔW=alpha*(vh'-v'h')
visbiasinc = momentum*visbiasinc + (epsilonvb/numcases)*(posvisact-negvisact); %可視層增量 Δa=alpha*(v-v')
hidbiasinc = momentum*hidbiasinc + (epsilonhb/numcases)*(poshidact-neghidact); %隱含層增量 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. 玻爾茲曼機與受限玻爾茲曼機
6. 參考文獻
[1] 邱錫鵬, 神經網絡與深度學習[M]. 2019.
[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.
[4] Hinton G E. Training products of experts by minimizing contrastive divergence[J]. Neural computation, 2002, 14(8): 1771-1800.
[5] Hinton G E. A practical guide to training restricted Boltzmann machines[M]//Neural networks: Tricks of the trade. Springer, Berlin, Heidelberg, 2012: 599-619.