一、softmax
softmax模型的含义是假设后验概率P(y|x)服从多项式分布，y=1,2,3,4,..,k,即有k类，根据多项式分布(n=1，也可以称为目录分布)的定义：
二、从广义线性模型中推导出softmax模型
我们的目标是给定X，求出参数phi，需要建立参数phi对X的模型，下面给出模型的推导。
下面我们将后验概率写成指数函数族的形式，以得出
三、优化函数与梯度
现在我们已经建立了参数phi对X的模型，下面需要做的是估计参数theta的值，利用最大似然估计即可。
下面求解梯度：
四、正则惩罚
为了使目标函数严格凸函数即存在唯一最小值，再加入一个权值惩罚项，得到新的目标函数与梯度：
五、matlab实验
实验数据用到了mnist数据库，用于识别10个手写数字。
%% 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....