基于自动编码机（autoencoder）,这里网络的层次结构为一个输入层，两个隐层，后面再跟着一个softmax分类器:
采用贪婪算法，首先把input和feature1看作一个自动编码机，训练出二者之间的参数，然后用feature1层的激活值作为输出，输入到feature2,即把feature1和feature2再看作一个自动编码机，训练出这两层之间的参数，这两步都没有用到分类标签，所以是无监督学习，最后把feature2的激活值作为提取的的特征，输入到分类器，这里需要标签来计算代价函数，从而由优化这个代价函数来训练出feature2与分类器之间的参数，所以这一步是有监督学习，这一步完成之后，把测试样本输入网络，最后会输出该样本分别属于每一类的概率，选出最大概率对应的类别，就是最终的分类结果。
为了使得分类结果更加精确，可以对训练出的参数进行微调，就是在有监督学习之后，我们利用有标签的训练数据可以计算出分类残差，然后利用这个残差反向传播，对已经训练出的参数进行进一步微调，会对最终预测的精度有很大提升
下面是第一层训学习出的特征：
可以看出都是一些笔迹的边缘
作为对比，训练结果显示，微调之后，分类准确度有大幅提升，所以在训练深度网络之后，利用部分标签数据进行微调是一件很有必要的学习：
Before Finetuning Test Accuracy: 91.760%
After Finetuning Test Accuracy: 97.710%
下面是部分程序代码，需要用到，完整代码请先下载minFunc.rar，然后下载stacked_exercise.rar，minFunc.rar里面是lbfgs优化函数，在优化网络参数时需要用到。
%% CS294A/CS294W Stacked Autoencoder Exercise
% Instructions
% ------------
%
% This file contains code that helps you get started on the
% sstacked autoencoder exercise. You will need to complete code in
% stackedAECost.m
% You will also need to have implemented sparseAutoencoderCost.m and
% softmaxCost.m from previous exercises. You will need the initializeParameters.m
% loadMNISTImages.m, and loadMNISTLabels.m files from previous exercises.
%
% For the purpose of completing the assignment, you do not need to
% change the code in this file.
%
%%======================================================================
%% STEP 0: Here we provide the relevant parameters values that will
% allow your sparse autoencoder to get good filters; you do not need to
% change the parameters below.
inputSize = 28 * 28;
numClasses = 10;
hiddenSizeL1 = 200; % Layer 1 Hidden Size
hiddenSizeL2 = 200; % Layer 2 Hidden Size
sparsityParam = 0.1; % desired average activation of the hidden units.
% (This was denoted by the Greek alphabet rho, which looks like a lower-case "p",
% in the lecture notes).
lambda = 3e-3; % weight decay parameter
beta = 3; % weight of sparsity penalty term
%%======================================================================
%% STEP 1: Load data from the MNIST database
%
% This loads our training data from the MNIST database files.
% Load MNIST database files
trainData = loadMNISTImages('train-images.idx3-ubyte');
trainLabels = loadMNISTLabels('train-labels.idx1-ubyte');
trainLabels(trainLabels == 0) = 10; % Remap 0 to 10 since our labels need to start from 1
%%======================================================================
%% STEP 2: Train the first sparse autoencoder
% This trains the first sparse autoencoder on the unlabelled STL training
% images.
% If you've correctly implemented sparseAutoencoderCost.m, you don't need
% to change anything here.
% Randomly initialize the parameters
sae1Theta = initializeParameters(hiddenSizeL1, inputSize);
%% ---------------------- YOUR CODE HERE ---------------------------------
% Instructions: Train the first layer sparse autoencoder, this layer has
% an hidden size of "hiddenSizeL1"
% You should store the optimal parameters in sae1OptTheta
addpath minFunc/;
options = struct;
options.Method = 'lbfgs';
options.maxIter = 400;
options.display = 'on';
%训练出第一层网络的参数
[sae1OptTheta, cost] = minFunc(@(p) sparseAutoencoderCost(p,...
inputSize,hiddenSizeL1,lambda,...
sparsityParam,beta,trainData),...
sae1Theta,options);
save('step2.mat', 'sae1OptTheta');
W1 = reshape(sae1OptTheta(1:hiddenSizeL1 * inputSize), hiddenSizeL1, inputSize);
display_network(W1');
% -------------------------------------------------------------------------
%%======================================================================
%% STEP 2: Train the second sparse autoencoder
% This trains the second sparse autoencoder on the first autoencoder
% featurse.
% If you've correctly implemented sparseAutoencoderCost.m, you don't need
% to change anything here.
[sae1Features] = feedForwardAutoencoder(sae1OptTheta, hiddenSizeL1, ...
inputSize, trainData);
% Randomly initialize the parameters
sae2Theta = initializeParameters(hiddenSizeL2, hiddenSizeL1);
%% ---------------------- YOUR CODE HERE ---------------------------------
% Instructions: Train the second layer sparse autoencoder, this layer has
% an hidden size of "hiddenSizeL2" and an inputsize of
% "hiddenSizeL1"
%
% You should store the optimal parameters in sae2OptTheta
[sae2OptTheta, cost] = minFunc(@(p)sparseAutoencoderCost(p,...
hiddenSizeL1,hiddenSizeL2,lambda,...
sparsityParam,beta,sae1Features),...
sae2Theta,options);
% figure;
% W11 = reshape(sae1OptTheta(1:hiddenSizeL1 * inputSize), hiddenSizeL1, inputSize);
% W2 = reshape(sae2OptTheta(1:hiddenSizeL2 * hiddenSizeL1), hiddenSizeL2, hiddenSizeL1);
% figure;
% display_network(W2');
% -------------------------------------------------------------------------
%%======================================================================
%% STEP 3: Train the softmax classifier
% This trains the sparse autoencoder on the second autoencoder features.
% If you've correctly implemented softmaxCost.m, you don't need
% to change anything here.
[sae2Features] = feedForwardAutoencoder(sae2OptTheta, hiddenSizeL2, ...
hiddenSizeL1, sae1Features);
% Randomly initialize the parameters
saeSoftmaxTheta = 0.005 * randn(hiddenSizeL2 * numClasses, 1);
%% ---------------------- YOUR CODE HERE ---------------------------------
% Instructions: Train the softmax classifier, the classifier takes in
% input of dimension "hiddenSizeL2" corresponding to the
% hidden layer size of the 2nd layer.
%
% You should store the optimal parameters in saeSoftmaxOptTheta
%
% NOTE: If you used softmaxTrain to complete this part of the exercise,
% set saeSoftmaxOptTheta = softmaxModel.optTheta(:);
softmaxLambda = 1e-4;
numClasses = 10;
softoptions = struct;
softoptions.maxIter = 400;
softmaxModel = softmaxTrain(hiddenSizeL2,numClasses,softmaxLambda,...
sae2Features,trainLabels,softoptions);
saeSoftmaxOptTheta = softmaxModel.optTheta(:);
save('step4.mat', 'saeSoftmaxOptTheta');
% -------------------------------------------------------------------------
%%======================================================================
%% STEP 5: Finetune softmax model
% Implement the stackedAECost to give the combined cost of the whole model
% then run this cell.
% Initialize the stack using the parameters learned
stack = cell(2,1);
stack{1}.w = reshape(sae1OptTheta(1:hiddenSizeL1*inputSize), ...
hiddenSizeL1, inputSize);
stack{1}.b = sae1OptTheta(2*hiddenSizeL1*inputSize+1:2*hiddenSizeL1*inputSize+hiddenSizeL1);
stack{2}.w = reshape(sae2OptTheta(1:hiddenSizeL2*hiddenSizeL1), ...
hiddenSizeL2, hiddenSizeL1);
stack{2}.b = sae2OptTheta(2*hiddenSizeL2*hiddenSizeL1+1:2*hiddenSizeL2*hiddenSizeL1+hiddenSizeL2);
% Initialize the parameters for the deep model
[stackparams, netconfig] = stack2params(stack);
stackedAETheta = [ saeSoftmaxOptTheta ; stackparams ];
%% ---------------------- YOUR CODE HERE ---------------------------------
% Instructions: Train the deep network, hidden size here refers to the '
% dimension of the input to the classifier, which corresponds
% to "hiddenSizeL2".
%
%
[stackedAEOptTheta, cost] = minFunc(@(p)stackedAECost(p,inputSize,hiddenSizeL2,...
numClasses, netconfig,lambda, trainData, trainLabels),...
stackedAETheta,options);
save('step5.mat', 'stackedAEOptTheta');
% -------------------------------------------------------------------------
%%======================================================================
%% STEP 6: Test
% Instructions: You will need to complete the code in stackedAEPredict.m
% before running this part of the code
%
% Get labelled test images
% Note that we apply the same kind of preprocessing as the training set
testData = loadMNISTImages('t10k-images.idx3-ubyte');
testLabels = loadMNISTLabels('t10k-labels.idx1-ubyte');
testLabels(testLabels == 0) = 10; % Remap 0 to 10
[pred] = stackedAEPredict(stackedAETheta, inputSize, hiddenSizeL2, ...
numClasses, netconfig, testData);
acc = mean(testLabels(:) == pred(:));
fprintf('Before Finetuning Test Accuracy: %0.3f%%
', acc * 100);
[pred] = stackedAEPredict(stackedAEOptTheta, inputSize, hiddenSizeL2, ...
numClasses, netconfig, testData);
acc = mean(testLabels(:) == pred(:));
fprintf('After Finetuning Test Accuracy: %0.3f%%
', acc * 100);
% Accuracy is the proportion of correctly classified images
% The results for our implementation were:
%
% Before Finetuning Test Accuracy: 87.7%
% After Finetuning Test Accuracy: 97.6%
%
% If your values are too low (accuracy less than 95%), you should check
% your code for errors, and make sure you are training on the
% entire data set of 60000 28x28 training images
% (unless you modified the loading code, this should be the case)
function [ cost, grad ] = stackedAECost(theta, inputSize, hiddenSize, ...
numClasses, netconfig, ...
lambda, data, labels)
% stackedAECost: Takes a trained softmaxTheta and a training data set with labels,
% and returns cost and gradient using a stacked autoencoder model. Used for
% finetuning.
% theta: trained weights from the autoencoder
% visibleSize: the number of input units
% hiddenSize: the number of hidden units *at the 2nd layer*
% numClasses: the number of categories
% netconfig: the network configuration of the stack
% lambda: the weight regularization penalty
% data: Our matrix containing the training data as columns. So, data(:,i) is the i-th training example.
% labels: A vector containing labels, where labels(i) is the label for the
% i-th training example
%% Unroll softmaxTheta parameter
% We first extract the part which compute the softmax gradient
softmaxTheta = reshape(theta(1:hiddenSize*numClasses), numClasses, hiddenSize);
% Extract out the "stack"
stack = params2stack(theta(hiddenSize*numClasses+1:end), netconfig);
% You will need to compute the following gradients
softmaxThetaGrad = zeros(size(softmaxTheta));
stackgrad = cell(size(stack));
for d = 1:numel(stack)
stackgrad{d}.w = zeros(size(stack{d}.w));
stackgrad{d}.b = zeros(size(stack{d}.b));
end
cost = 0; % You need to compute this
% You might find these variables useful
M = size(data, 2);
groundTruth = full(sparse(labels, 1:M, 1));
%% --------------------------- YOUR CODE HERE -----------------------------
% Instructions: Compute the cost function and gradient vector for
% the stacked autoencoder.
%
% You are given a stack variable which is a cell-array of
% the weights and biases for every layer. In particular, you
% can refer to the weights of Layer d, using stack{d}.w and
% the biases using stack{d}.b . To get the total number of
% layers, you can use numel(stack).
%
% The last layer of the network is connected to the softmax
% classification layer, softmaxTheta.
%
% You should compute the gradients for the softmaxTheta,
% storing that in softmaxThetaGrad. Similarly, you should
% compute the gradients for each layer in the stack, storing
% the gradients in stackgrad{d}.w and stackgrad{d}.b
% Note that the size of the matrices in stackgrad should
% match exactly that of the size of the matrices in stack.
%
depth = numel(stack);
z = cell(depth+1,1);
a = cell(depth+1, 1);
a{1} = data;
for layer = (1:depth)
z{layer+1} = stack{layer}.w * a{layer} + repmat(stack{layer}.b, [1, size(a{layer},2)]);
a{layer+1} = sigmoid(z{layer+1});
end
M = softmaxTheta * a{depth+1};
M = bsxfun(@minus, M, max(M));
p = bsxfun(@rdivide, exp(M), sum(exp(M)));
cost = -1/numClasses * groundTruth(:)' * log(p(:)) + lambda/2 * sum(softmaxTheta(:) .^ 2);
softmaxThetaGrad = -1/numClasses * (groundTruth - p) * a{depth+1}' + lambda * softmaxTheta;
d = cell(depth+1);
d{depth+1} = -(softmaxTheta' * (groundTruth - p)) .* a{depth+1} .* (1-a{depth+1});
for layer = (depth:-1:2)
d{layer} = (stack{layer}.w' * d{layer+1}) .* a{layer} .* (1-a{layer});
end
for layer = (depth:-1:1)
stackgrad{layer}.w = (1/numClasses) * d{layer+1} * a{layer}';
stackgrad{layer}.b = (1/numClasses) * sum(d{layer+1}, 2);
end
% -------------------------------------------------------------------------
%% Roll gradient vector
grad = [softmaxThetaGrad(:) ; stack2params(stackgrad)];
end
% You might find this useful
function sigm = sigmoid(x)
sigm = 1 ./ (1 + exp(-x));
end
function [pred] = stackedAEPredict(theta, inputSize, hiddenSize, numClasses, netconfig, data)
% stackedAEPredict: Takes a trained theta and a test data set,
% and returns the predicted labels for each example.
% theta: trained weights from the autoencoder
% visibleSize: the number of input units
% hiddenSize: the number of hidden units *at the 2nd layer*
% numClasses: the number of categories
% data: Our matrix containing the training data as columns. So, data(:,i) is the i-th training example.
% Your code should produce the prediction matrix
% pred, where pred(i) is argmax_c P(y(c) | x(i)).
%% Unroll theta parameter
% We first extract the part which compute the softmax gradient
softmaxTheta = reshape(theta(1:hiddenSize*numClasses), numClasses, hiddenSize);
% Extract out the "stack"
stack = params2stack(theta(hiddenSize*numClasses+1:end), netconfig);
%% ---------- YOUR CODE HERE --------------------------------------
% Instructions: Compute pred using theta assuming that the labels start
% from 1.
depth = numel(stack);
z = cell(depth+1,1);
a = cell(depth+1, 1);
a{1} = data;
for layer = (1:depth)
z{layer+1} = stack{layer}.w * a{layer} + repmat(stack{layer}.b, [1, size(a{layer},2)]);
a{layer+1} = sigmoid(z{layer+1});
end
[~, pred] = max(softmaxTheta * a{depth+1});
% -----------------------------------------------------------
end
% You might find this useful
function sigm = sigmoid(x)
sigm = 1 ./ (1 + exp(-x));
end