Deep Learning 學習隨記（六）Linear Decoder 線性解碼

線性解碼器（Linear Decoder）

前面第一章提到稀疏自編碼器（http://www.cnblogs.com/bzjia-blog/p/SparseAutoencoder.html）的三層網絡結構，咱們要知足最後一層的輸出：a^（3）≈a^（1）（即輸入值x）的近似重建。考慮到在最後一層的a^（3）=f（z^（3）），這裏f通常用sigmoid函數或tanh函數等非線性函數，而將輸出界定在一個範圍內（好比sigmoid函數使結果在[0，1]中）。這對於有些數據組，例如MNIST手寫數字庫中其輸入輸出範圍符合極佳，但並非全部的狀況都知足這個條件。例如，若採用PCA白化，輸入將再也不限制於[0,1]，雖可經過縮放數據來確保其符合特定範圍內，但顯然，這不是最好的方式。

所以，這裏提到的Linear Decoder就是經過在最後一層用激勵函數：a^（3） = z^（3）（也即f（z）=z）來實現。這裏要注意到，只是在最後一層用這個激勵函數，其餘隱層的激勵函數仍然是sigmoid函數或者tanh函數，咱們僅在輸出層中使用線性激勵機制。

這樣一來，在求梯度的時候，公式：

 

就應該改爲：

這個是顯然的，由於f'（z）=1。其餘層的都不須要改變。

 

練習：

這裏講義給出了一個練習，基本跟稀疏自編碼同樣，只有幾處須要稍微改動一下。

linearDecoderExercise.m

%% CS294A/CS294W Linear Decoder Exercise

%  Instructions
%  ------------
% 
%  This file contains code that helps you get started on the
%  linear decoder exericse. For this exercise, you will only need to modify
%  the code in sparseAutoencoderLinearCost.m. You will not need to modify
%  any code in this file.

%%======================================================================
%% STEP 0: Initialization
%  Here we initialize some parameters used for the exercise.

imageChannels = 3;     % number of channels (rgb, so 3)

patchDim   = 8;          % patch dimension
numPatches = 100000;   % number of patches

visibleSize = patchDim * patchDim * imageChannels;  % number of input units 
outputSize  = visibleSize;   % number of output units
hiddenSize  = 400;           % number of hidden units 

sparsityParam = 0.035; % desired average activation of the hidden units.
lambda = 3e-3;         % weight decay parameter       
beta = 5;              % weight of sparsity penalty term       

epsilon = 0.1;           % epsilon for ZCA whitening

%%======================================================================
%% STEP 1: Create and modify sparseAutoencoderLinearCost.m to use a linear decoder,
%          and check gradients
%  You should copy sparseAutoencoderCost.m from your earlier exercise 
%  and rename it to sparseAutoencoderLinearCost.m. 
%  Then you need to rename the function from sparseAutoencoderCost to
%  sparseAutoencoderLinearCost, and modify it so that the sparse autoencoder
%  uses a linear decoder instead. Once that is done, you should check 
% your gradients to verify that they are correct.

% NOTE: Modify sparseAutoencoderCost first!

% To speed up gradient checking, we will use a reduced network and some
% dummy patches

debugHiddenSize = 5;
debugvisibleSize = 8;
patches = rand([8 10]);
theta = initializeParameters(debugHiddenSize, debugvisibleSize); 

[cost, grad] = sparseAutoencoderLinearCost(theta, debugvisibleSize, debugHiddenSize, ...
                                           lambda, sparsityParam, beta, ...
                                           patches);

% Check gradients
numGrad = computeNumericalGradient( @(x) sparseAutoencoderLinearCost(x, debugvisibleSize, debugHiddenSize, ...
                                                  lambda, sparsityParam, beta, ...
                                                  patches), theta);

% Use this to visually compare the gradients side by side
disp([numGrad grad]); 

diff = norm(numGrad-grad)/norm(numGrad+grad);
% Should be small. In our implementation, these values are usually less than 1e-9.
disp(diff); 

assert(diff < 1e-9, 'Difference too large. Check your gradient computation again');

% NOTE: Once your gradients check out, you should run step 0 again to
%       reinitialize the parameters
%}

%%======================================================================
%% STEP 2: Learn features on small patches
%  In this step, you will use your sparse autoencoder (which now uses a 
%  linear decoder) to learn features on small patches sampled from related
%  images.

%% STEP 2a: Load patches
%  In this step, we load 100k patches sampled from the STL10 dataset and
%  visualize them. Note that these patches have been scaled to [0,1]

load stlSampledPatches.mat

displayColorNetwork(patches(:, 1:100));

%% STEP 2b: Apply preprocessing
%  In this sub-step, we preprocess the sampled patches, in particular, 
%  ZCA whitening them. 
% 
%  In a later exercise on convolution and pooling, you will need to replicate 
%  exactly the preprocessing steps you apply to these patches before 
%  using the autoencoder to learn features on them. Hence, we will save the
%  ZCA whitening and mean image matrices together with the learned features
%  later on.

% Subtract mean patch (hence zeroing the mean of the patches)
meanPatch = mean(patches, 2);  
patches = bsxfun(@minus, patches, meanPatch);

% Apply ZCA whitening
sigma = patches * patches' / numPatches;
[u, s, v] = svd(sigma);
ZCAWhite = u * diag(1 ./ sqrt(diag(s) + epsilon)) * u';
patches = ZCAWhite * patches;

displayColorNetwork(patches(:, 1:100));

%% STEP 2c: Learn features
%  You will now use your sparse autoencoder (with linear decoder) to learn
%  features on the preprocessed patches. This should take around 45 minutes.

theta = initializeParameters(hiddenSize, visibleSize);

% Use minFunc to minimize the function
addpath minFunc/

options = struct;
options.Method = 'lbfgs'; 
options.maxIter = 400;
options.display = 'on';

[optTheta, cost] = minFunc( @(p) sparseAutoencoderLinearCost(p, ...
                                   visibleSize, hiddenSize, ...
                                   lambda, sparsityParam, ...
                                   beta, patches), ...
                              theta, options);

% Save the learned features and the preprocessing matrices for use in 
% the later exercise on convolution and pooling
fprintf('Saving learned features and preprocessing matrices...\n');                          
save('STL10Features.mat', 'optTheta', 'ZCAWhite', 'meanPatch');
fprintf('Saved\n');

%% STEP 2d: Visualize learned features

W = reshape(optTheta(1:visibleSize * hiddenSize), hiddenSize, visibleSize);
b = optTheta(2*hiddenSize*visibleSize+1:2*hiddenSize*visibleSize+hiddenSize);
displayColorNetwork( (W*ZCAWhite)');

 

 sparseAutoencoderLinearCost.m

function [cost,grad,features] = sparseAutoencoderLinearCost(theta, visibleSize, hiddenSize, ...
                                                            lambda, sparsityParam, beta, data)
% -------------------- YOUR CODE HERE --------------------
% Instructions:
%   Copy sparseAutoencoderCost in sparseAutoencoderCost.m from your
%   earlier exercise onto this file, renaming the function to
%   sparseAutoencoderLinearCost, and changing the autoencoder to use a
%   linear decoder.
                                   
% visibleSize: the number of input units (probably 64) 
% hiddenSize: the number of hidden units (probably 25) 
% lambda: weight decay parameter
% sparsityParam: The desired average activation for the hidden units (denoted in the lecture
%                           notes by the greek alphabet rho, which looks like a lower-case "p").
% beta: weight of sparsity penalty term
% data: Our 64x10000 matrix containing the training data.  So, data(:,i) is the i-th training example. 
  
% The input theta is a vector (because minFunc expects the parameters to be a vector). 
% We first convert theta to the (W1, W2, b1, b2) matrix/vector format, so that this 
% follows the notation convention of the lecture notes. 

%將長向量轉換成每一層的權值矩陣和偏置向量值
W1 = reshape(theta(1:hiddenSize*visibleSize), hiddenSize, visibleSize);
W2 = reshape(theta(hiddenSize*visibleSize+1:2*hiddenSize*visibleSize), visibleSize, hiddenSize);
b1 = theta(2*hiddenSize*visibleSize+1:2*hiddenSize*visibleSize+hiddenSize);
b2 = theta(2*hiddenSize*visibleSize+hiddenSize+1:end);

% Cost and gradient variables (your code needs to compute these values). 
% Here, we initialize them to zeros. 
cost = 0;
W1grad = zeros(size(W1)); 
W2grad = zeros(size(W2));
b1grad = zeros(size(b1)); 
b2grad = zeros(size(b2));

%% ---------- YOUR CODE HERE --------------------------------------


Jcost = 0;%直接偏差
Jweight = 0;%權值懲罰
Jsparse = 0;%稀疏性懲罰
[n m] = size(data);%m爲樣本的個數，n爲樣本的特徵數

%前向算法計算各神經網絡節點的線性組合值和active值
z2 = W1*data+repmat(b1,1,m);%注意這裏必定要將b1向量複製擴展成m列的矩陣
a2 = sigmoid(z2);
z3 = W2*a2+repmat(b2,1,m);
a3 = z3;                                             %線性解碼器************

% 計算預測產生的偏差
Jcost = (0.5/m)*sum(sum((a3-data).^2));

%計算權值懲罰項
Jweight = (1/2)*(sum(sum(W1.^2))+sum(sum(W2.^2)));

%計算稀釋性規則項
rho = (1/m).*sum(a2,2);%求出第一個隱含層的平均值向量
Jsparse = sum(sparsityParam.*log(sparsityParam./rho)+ ...
        (1-sparsityParam).*log((1-sparsityParam)./(1-rho)));

%損失函數的總表達式
cost = Jcost+lambda*Jweight+beta*Jsparse;

%反向算法求出每一個節點的偏差值
d3 = -(data-a3);                                         %線性解碼器**************
sterm = beta*(-sparsityParam./rho+(1-sparsityParam)./(1-rho));%由於加入了稀疏規則項，因此
                                                             %計算偏導時須要引入該項
d2 = (W2'*d3+repmat(sterm,1,m)).*sigmoidInv(z2); 

%計算W1grad 
W1grad = W1grad+d2*data';
W1grad = (1/m)*W1grad+lambda*W1;

%計算W2grad  
W2grad = W2grad+d3*a2';
W2grad = (1/m).*W2grad+lambda*W2;

%計算b1grad 
b1grad = b1grad+sum(d2,2);
b1grad = (1/m)*b1grad;%注意b的偏導是一個向量，因此這裏應該把每一行的值累加起來

%計算b2grad 
b2grad = b2grad+sum(d3,2);
b2grad = (1/m)*b2grad;

%-------------------------------------------------------------------
% After computing the cost and gradient, we will convert the gradients back
% to a vector format (suitable for minFunc).  Specifically, we will unroll
% your gradient matrices into a vector.

grad = [W1grad(:) ; W2grad(:) ; b1grad(:) ; b2grad(:)];

end

%-------------------------------------------------------------------
% Here's an implementation of the sigmoid function, which you may find useful
% in your computation of the costs and the gradients.  This inputs a (row or
% column) vector (say (z1, z2, z3)) and returns (f(z1), f(z2), f(z3)). 

function sigm = sigmoid(x)

    sigm = 1 ./ (1 + exp(-x));
end

%sigmoid函數的逆函數
function sigmInv = sigmoidInv(x)

    sigmInv = sigmoid(x).*(1-sigmoid(x));
end

只是對稀疏自編碼器的代碼進行了兩處稍微的改動。

結果：

學習到的特徵也放在了STL10Features.mat裏，將要在下一章的練習中用到。

 PS:講義地址：

http://deeplearning.stanford.edu/wiki/index.php/Linear_Decoders

http://deeplearning.stanford.edu/wiki/index.php/Exercise:Learning_color_features_with_Sparse_Autoencoders