Softmax Regression練習

    在上篇博文(http://blog.csdn.net/freeliao/article/details/19424565)介紹了Softmax Regression的模型，現在來做下該模型在MNIST數據集上的識別練習(http://ufldl.stanford.edu/wiki/index.php/Exercise:Softmax_Regression)。MNIST數據集訓練集由60000張28*28的圖片組成，測試集由10000張同樣大小的圖片組成。

    這裏的Softmax Regression沒有隱含層，直接以圖片的像素值作爲輸入，得到輸出層，輸出層有10個單元，每個單元代表輸入屬於該類別的概率。

回顧一下模型的損失函數和梯度：

    matlab實現過程中要注意用Vectorized的實現方式，儘量避免出現構造過大的矩陣，否則容易出現out of memory錯誤。

  實現結果準確率爲92.640%。

softmaxExercise.m

%% CS294A/CS294W Softmax Exercise 

%  Instructions
%  ------------
% 
%  This file contains code that helps you get started on the
%  softmax exercise. You will need to write the softmax cost function 
%  in softmaxCost.m and the softmax prediction function in softmaxPred.m. 
%  For this exercise, you will not need to change any code in this file,
%  or any other files other than those mentioned above.
%  (However, you may be required to do so in later exercises)

%%======================================================================
%% STEP 0: Initialise constants and parameters
%
%  Here we define and initialise some constants which allow your code
%  to be used more generally on any arbitrary input. 
%  We also initialise some parameters used for tuning the model.

inputSize = 28 * 28; % Size of input vector (MNIST images are 28x28)
numClasses = 10;     % Number of classes (MNIST images fall into 10 classes)

lambda = 1e-4; % Weight decay parameter

%%======================================================================
%% STEP 1: Load data
%
%  In this section, we load the input and output data.
%  For softmax regression on MNIST pixels, 
%  the input data is the images, and 
%  the output data is the labels.
%

% Change the filenames if you've saved the files under different names
% On some platforms, the files might be saved as 
% train-images.idx3-ubyte / train-labels.idx1-ubyte

images = loadMNISTImages('mnist/train-images-idx3-ubyte');
labels = loadMNISTLabels('mnist/train-labels-idx1-ubyte');
labels(labels==0) = 10; % Remap 0 to 10

inputData = images;

% For debugging purposes, you may wish to reduce the size of the input data
% in order to speed up gradient checking. 
% Here, we create synthetic dataset using random data for testing

DEBUG = false; % Set DEBUG to true when debugging.
if DEBUG
    inputSize = 8;
    inputData = randn(8, 100);
    labels = randi(10, 100, 1);
end

% Randomly initialise theta
theta = 0.005 * randn(numClasses * inputSize, 1);

%%======================================================================
%% STEP 2: Implement softmaxCost
%
%  Implement softmaxCost in softmaxCost.m. 

[cost, grad] = softmaxCost(theta, numClasses, inputSize, lambda, inputData, labels);
                                     
%%======================================================================
%% STEP 3: Gradient checking
%
%  As with any learning algorithm, you should always check that your
%  gradients are correct before learning the parameters.
% 

if DEBUG
    numGrad = computeNumericalGradient( @(x) softmaxCost(x, numClasses, ...
                                    inputSize, lambda, inputData, labels), theta);

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

    % Compare numerically computed gradients with those computed analytically
    diff = norm(numGrad-grad)/norm(numGrad+grad);
    disp(diff); 
    % The difference should be small. 
    % In our implementation, these values are usually less than 1e-7.

    % When your gradients are correct, congratulations!
end

%%======================================================================
%% STEP 4: Learning parameters
%
%  Once you have verified that your gradients are correct, 
%  you can start training your softmax regression code using softmaxTrain
%  (which uses minFunc).

options.maxIter = 100;
softmaxModel = softmaxTrain(inputSize, numClasses, lambda, ...
                            inputData, labels, options);
                          
% Although we only use 100 iterations here to train a classifier for the 
% MNIST data set, in practice, training for more iterations is usually
% beneficial.

%%======================================================================
%% STEP 5: Testing
%
%  You should now test your model against the test images.
%  To do this, you will first need to write softmaxPredict
%  (in softmaxPredict.m), which should return predictions
%  given a softmax model and the input data.

images = loadMNISTImages('mnist/t10k-images-idx3-ubyte');
labels = loadMNISTLabels('mnist/t10k-labels-idx1-ubyte');
labels(labels==0) = 10; % Remap 0 to 10

inputData = images;

% You will have to implement softmaxPredict in softmaxPredict.m
[pred] = softmaxPredict(softmaxModel, inputData);

acc = mean(labels(:) == pred(:));
fprintf('Accuracy: %0.3f%%\n', acc * 100);

% Accuracy is the proportion of correctly classified images
% After 100 iterations, the results for our implementation were:
%
% Accuracy: 92.200%
%
% If your values are too low (accuracy less than 0.91), 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)

softmaxCost.m

function [cost, grad] = softmaxCost(theta, numClasses, inputSize, lambda, data, labels)

% numClasses - the number of classes 
% inputSize - the size N of the input vector
% lambda - weight decay parameter
% data - the N x M input matrix, where each column data(:, i) corresponds to
%        a single test set
% labels - an M x 1 matrix containing the labels corresponding for the input data
%

% Unroll the parameters from theta
theta = reshape(theta, numClasses, inputSize);

numCases = size(data, 2);%樣本數
%sparse(labels,1:numCases,1)構建稀疏矩陣，labels爲行號，1:numCases爲列號
%full(*)還原稀疏矩陣，得到的矩陣每一列爲一個樣本的預測值向量
groundTruth = full(sparse(labels, 1:numCases, 1)); 
cost = 0;

thetagrad = zeros(numClasses, inputSize);

%% ---------- YOUR CODE HERE --------------------------------------
%  Instructions: Compute the cost and gradient for softmax regression.
%                You need to compute thetagrad and cost.
%                The groundTruth matrix might come in handy.
M = theta *data; %M的每一列爲一個樣本屬於1:numClasses的theta'*x
M = bsxfun(@minus, M, max(M, [], 1)); %減去每一列的最大值以避免溢出
M = exp(M);
p = bsxfun(@rdivide, M, sum(M)); %得到概率矩陣

%之前天真地用 -1/numCases .* sum(diag(groundTruth'*log(p))) + lambda/2 * sum(theta(:).^2)
%去算cost造成out of memory，以爲是內存不夠，真不應該啊
cost = -1/numCases .* sum(groundTruth(:)'*log(p(:))) + lambda/2 * sum(theta(:).^2);
thetagrad = -1/numCases .* (groundTruth - p) * data' + lambda * theta;



% ------------------------------------------------------------------
% Unroll the gradient matrices into a vector for minFunc
grad = [thetagrad(:)];
end

softmaxTrain.m

function [softmaxModel] = softmaxTrain(inputSize, numClasses, lambda, inputData, labels, options)
%softmaxTrain Train a softmax model with the given parameters on the given
% data. Returns softmaxOptTheta, a vector containing the trained parameters
% for the model.
%
% inputSize: the size of an input vector x^(i)
% numClasses: the number of classes 
% lambda: weight decay parameter
% inputData: an N by M matrix containing the input data, such that
%            inputData(:, c) is the cth input
% labels: M by 1 matrix containing the class labels for the
%            corresponding inputs. labels(c) is the class label for
%            the cth input
% options (optional): options
%   options.maxIter: number of iterations to train for

if ~exist('options', 'var')
    options = struct;
end

if ~isfield(options, 'maxIter')
    options.maxIter = 400;
end

% initialize parameters
theta = 0.005 * randn(numClasses * inputSize, 1);

% Use minFunc to minimize the function
addpath 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,
                          % softmaxCost.m satisfies this.
minFuncOptions.display = 'on';

[softmaxOptTheta, cost] = minFunc( @(p) softmaxCost(p, ...
                                   numClasses, inputSize, lambda, ...
                                   inputData, labels), ...                                   
                              theta, options);

% Fold softmaxOptTheta into a nicer format
softmaxModel.optTheta = reshape(softmaxOptTheta, numClasses, inputSize);
softmaxModel.inputSize = inputSize;
softmaxModel.numClasses = numClasses;
                          
end                          

softmaxPredict.m

function [pred] = softmaxPredict(softmaxModel, data)

% softmaxModel - model trained using softmaxTrain
% data - the N x M input matrix, where each column data(:, i) corresponds to
%        a single test set
%
% Your code should produce the prediction matrix 
% pred, where pred(i) is argmax_c P(y(c) | x(i)).
 
% Unroll the parameters from theta
theta = softmaxModel.optTheta;  % this provides a numClasses x inputSize matrix
pred = zeros(1, size(data, 2));

%% ---------- YOUR CODE HERE --------------------------------------
%  Instructions: Compute pred using theta assuming that the labels start 
%                from 1.
%prob爲矩陣每列最大概率值，pred爲該列行號索引
[prob pred] = max(theta*data);




% ---------------------------------------------------------------------

end