深度学习基础（五）Softmax Regression

Softmax Regression

Softmax Regression是 Logistic Regression的推广

假设我们有训练集


Logistic Regression:

对于每个特征

，标签








Softmax Regression:

对于每个特征

，标签






Softmax Regression有一个很特别地性质：过参数化



可以看到参数减去任意的一个值并不影响我们的假设，也就是说有很多歌参数满足我们的假设

为了避免过大参数的影响，





参考练习：
http://deeplearning.stanford.edu/wiki/index.php/Exercise:Softmax_Regression
实验步骤：

0. 初始化参数和常亮

1.载入数据

2.计算代价函数

3.Gradient checking

4.训练

5.测试

%% 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('train-images-idx3-ubyte');
labels = loadMNISTLabels('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.

M = bsxfun(@minus, theta*data,max((theta*data),[],1));
M = exp(M);
p = bsxfun(@rdivide, M, sum(M));
cost = -1/numCases * groundTruth(:)'*log(p(:)) + lamda/2 * sum(theta(:)).^2;
thetagrad = -1/numCases * (groundTruth - p) *data' + lamda*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 forif
~exist('options', 'var') options = struct;endif ~isfield(options, 'maxIter') options.maxIter = 400;end% initialize parameterstheta = 0.005 * randn(numClasses * inputSize, 1);% 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, % 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 formatsoftmaxModel.optTheta = reshape(softmaxOptTheta, numClasses, inputSize);softmaxModel.inputSize = inputSize;softmaxModel.numClasses
= numClasses; 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 forif ~exist('options', 'var') options = struct;endif ~isfield(options, 'maxIter') options.maxIter = 400;end% initialize parameterstheta = 0.005 * randn(numClasses * inputSize, 1);% 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, % 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 formatsoftmaxModel.optTheta = reshape(softmaxOptTheta, numClasses, inputSize);softmaxModel.inputSize
= inputSize;softmaxModel.numClasses = numClasses; end