softmax原理及Matlab實現

一、softmax 

softmax模型的含義是假設後驗概率P(y|x)服從多項式分佈，y=1,2,3,4,..,k,即有k類，根據多項式分佈(n=1，也可以稱爲目錄分佈)的定義：

二、從廣義線性模型中推導出softmax模型

我們的目標是給定X，求出參數phi，需要建立參數phi對X的模型，下面給出模型的推導。

下面我們將後驗概率寫成指數函數族的形式，以得出

三、優化函數與梯度

現在我們已經建立了參數phi對X的模型，下面需要做的是估計參數theta的值，利用最大似然估計即可。

下面求解梯度：

四、正則懲罰

爲了使目標函數嚴格凸函數即存在唯一最小值，再加入一個權值懲罰項，得到新的目標函數與梯度：

五、matlab實驗

實驗數據用到了mnist數據庫，用於識別10個手寫數字。

%% 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 = true; % 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)

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);

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.
[N,M]=size(data);
eta=bsxfun(@minus,theta*data,max(theta*data,[],1));
eta=exp(eta);
pij=bsxfun(@rdivide,eta,sum(eta));

cost=-1./M*sum(sum(groundTruth.*log(pij)))+lambda/2*sum(sum(theta.^2));

thetagrad=-1/M.*(groundTruth-pij)*data'+lambda.*theta;
% ------------------------------------------------------------------
% Unroll the gradient matrices into a vector for minFunc
grad = [thetagrad(:)];
end

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                          

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 
[prob,pred]=max(theta*data);
% ---------------------------------------------------------------------
end

to be continued....