Convolutional neural networks(CNN) (九) Implement deep networks for digit classification Exercise

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{作为CNN学习入门的一部分，笔者在这里逐步给出UFLDL的各章节Exercise的个人代码实现，供大家参考指正}

理论部分可以在线参阅(页面最下方有中文选项)Self-Taught Learning to Deep Networks到Fine-tuning Stacked AEs部分内容

但大部分推导都可以在在线参阅的内容Fine-tuning Stacked AEs中找到，这里只给出对输出层error term求导的证明：

stackedAeCost.m

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 gradientsoftmaxTheta = 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 gradientssoftmaxThetaGrad = 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 usefulnumCases = size(data, 2);groundTruth = full(sparse(labels, 1:numCases, 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.%                %                Gradient Check : 1.0932e-11%%  ================= Cost =================%  Stack Layer activation_L1 = 1 ./ (1 + exp(-bsxfun(@plus, stack{1}.w*data, stack{1}.b)));activation_L2 = 1 ./ (1 + exp(-bsxfun(@plus, stack{2}.w*activation_L1, stack{2}.b)));sae2Features = activation_L2;%  Softmax LayerM = softmaxTheta * sae2Features; % M(r,c) is theta.T.r*x(:,c)M = bsxfun(@minus, M, max(M, [], 1)); % Preventing overflows.M = exp(M); M = bsxfun(@rdivide, M, sum(M)); % Dividing all elements in each column by their column sum.J_theta = sum(sum(log(M).*groundTruth));J_theta = -J_theta / numCases;WeightDecay = lambda * sum(sum(softmaxTheta.^2)) / 2;cost = J_theta + WeightDecay;%  ================= Gradient =================%  Stack LayerM = groundTruth - M;Gradient_J = softmaxTheta' * (M);MatrixDelte_L2 = -Gradient_J/numCases .* activation_L2 .* (1 - activation_L2);MatrixDelte_L1 = ((stack{2}.w)' * MatrixDelte_L2) .* activation_L1 .* (1 - activation_L1);stackgrad{1}.w = MatrixDelte_L1 * data';stackgrad{1}.b = sum(MatrixDelte_L1, 2);stackgrad{2}.w = MatrixDelte_L2 * activation_L1';stackgrad{2}.b = sum(MatrixDelte_L2, 2);%  Softmax Layerfor i = 1:1:numClasses    softmaxThetaGrad(i,:) = sum(bsxfun(@times, activation_L2, M(i,:)), 2); % Array multiplyendsoftmaxThetaGrad = -softmaxThetaGrad/numCases + lambda * softmaxTheta;% -------------------------------------------------------------------------%% Roll gradient vectorgrad = [softmaxThetaGrad(:) ; stack2params(stackgrad)];end

stackedAEExercise.m

%% 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 SizehiddenSizeL2 = 200;    % Layer 2 Hidden SizesparsityParam = 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    magnitude = 100;         % magnitude of iteration%%======================================================================%% STEP 1: Load data from the MNIST database%%  This loads our training data from the MNIST database files.%  Load MNIST database filestrainData = loadMNISTImages('mnist/train-images.idx3-ubyte');trainLabels = loadMNISTLabels('mnist/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 parameterssae1Theta = 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 sae1OptThetapatches = trainData;%  Use minFunc to minimize the functionaddpath minFunc/options.Method = 'lbfgs'; % Here, we use L-BFGS to optimize our cost                          % function. Generally, for minFunc to work, you                          % need a function pointer with two outputs: the                          % function value and the gradient. In our problem,                          % sparseAutoencoderCost.m satisfies this.options.maxIter = 4 * magnitude;  % Maximum number of iterations of L-BFGS to run options.display = 'on';[sae1OptTheta, cost] = minFunc( @(p) sparseAutoencoderCost(p, ...                                   inputSize, hiddenSizeL1, ...                                   lambda, sparsityParam, ...                                   beta, patches), ...                              sae1Theta, options);% -------------------------------------------------------------------------%%======================================================================%% 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 parameterssae2Theta = 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 sae2OptThetapatches = sae1Features;%  Use minFunc to minimize the functionaddpath minFunc/options.Method = 'lbfgs'; % Here, we use L-BFGS to optimize our cost                          % function. Generally, for minFunc to work, you                          % need a function pointer with two outputs: the                          % function value and the gradient. In our problem,                          % sparseAutoencoderCost.m satisfies this.options.maxIter = 4 * magnitude;  % Maximum number of iterations of L-BFGS to run options.display = 'on';[sae2OptTheta, cost] = minFunc( @(p) sparseAutoencoderCost(p, ...                                   hiddenSizeL1, hiddenSizeL2, ...                                   lambda, sparsityParam, ...                                   beta, patches), ...                              sae2Theta, options);% -------------------------------------------------------------------------%%======================================================================%% 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 parameterssaeSoftmaxTheta = 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(:);lambda = 1e-4;options.maxIter = 1 * magnitude;softmaxModel = softmaxTrain(hiddenSizeL2, numClasses, lambda, ...                            sae2Features, trainLabels, options);saeSoftmaxOptTheta = softmaxModel.optTheta(:);% -------------------------------------------------------------------------%%======================================================================%% 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 learnedstack = 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);   % -------------------------------------------------------------------------%%======================================================================%% 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 settestData = loadMNISTImages('mnist/t10k-images.idx3-ubyte');testLabels = loadMNISTLabels('mnist/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%%\n', acc * 100);[pred] = stackedAEPredict(stackedAEOptTheta, inputSize, hiddenSizeL2, ...                          numClasses, netconfig, testData);acc = mean(testLabels(:) == pred(:));fprintf('After Finetuning Test Accuracy: %0.3f%%\n', 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)

stackedAEPredict.m

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 gradientsoftmaxTheta = 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.activation_L1 = 1 ./ (1 + exp(-bsxfun(@plus, stack{1}.w*data, stack{1}.b)));activation_L2 = 1 ./ (1 + exp(-bsxfun(@plus, stack{2}.w*activation_L1, stack{2}.b)));sae2Features = activation_L2;M = softmaxTheta * sae2Features;[argmax_c_value_Vec, argmax_c_index_Vec] = max(M, [], 1);pred = argmax_c_index_Vec;% -----------------------------------------------------------end

实验结果：

笔者在实验中采用的迭代轮数为：

两个Sparse AutoEncoder Layer 迭代400次

Softmax Layer 迭代100次

Fine-tunning 迭代100次

权重损失系数：

AutoEncoder Layer 3e-3
Softmax Layer 1e-4

Fine-tunning 1e-4

可能由于系数选择的不同以及迭代次数的差异，笔者的实验结果与benchmark有较大差异：

Benchmark为：