深度学习 Deep LearningUFLDL 最新Tutorial 学习笔记 2:Logistic Regression
1 Logistic Regression 简述
Linear Regression 研究连续量的变化情况,而Logistic Regression则研究离散量的情况。简单地说就是对于推断一个训练样本是属于1还是0。那么非常easy地我们会想到概率,对,就是我们计算样本属于1的概率及属于0的概率,这样就能够依据概率来预计样本的情况,通过概率也将离散问题变成了连续问题。
[p]Specifically, we will try to learn a function of the form:
我们仅仅须要计算y=1的概率就ok了。其Cost Function例如以下:
J(θ)=−∑i(y(i)log(hθ(x(i)))+(1−y(i))log(1−hθ(x(i)))).
除了方程不一样,其它的计算和Linear Regression是全然一样的。
OK,接下来我们来看看练习怎么做。
2 exercise1B 解答
本练习通过使用MNIST的数据来推断手写数字0或者1.我直接贴出代码:ex1b_regression.m (无需更改)
addpath ../common addpath ../common/minFunc_2012/minFunc addpath ../common/minFunc_2012/minFunc/compiled % Load the MNIST data for this exercise. % train.X and test.X will contain the training and testing images. % Each matrix has size [n,m] where: % m is the number of examples. % n is the number of pixels in each image. % train.y and test.y will contain the corresponding labels (0 or 1). binary_digits = true; [train,test] = ex1_load_mnist(binary_digits); % Add row of 1s to the dataset to act as an intercept term. train.X = [ones(1,size(train.X,2)); train.X]; test.X = [ones(1,size(test.X,2)); test.X]; % Training set dimensions m=size(train.X,2); n=size(train.X,1); % Train logistic regression classifier using minFunc options = struct('MaxIter', 100); % First, we initialize theta to some small random values. theta = rand(n,1)*0.001; % Call minFunc with the logistic_regression.m file as the objective function. % % TODO: Implement batch logistic regression in the logistic_regression.m file! % %tic; %theta=minFunc(@logistic_regression, theta, options, train.X, train.y); %fprintf('Optimization took %f seconds.\n', toc); % Now, call minFunc again with logistic_regression_vec.m as objective. % % TODO: Implement batch logistic regression in logistic_regression_vec.m using % MATLAB's vectorization features to speed up your code. Compare the running % time for your logistic_regression.m and logistic_regression_vec.m implementations. % % Uncomment the lines below to run your vectorized code. %theta = rand(n,1)*0.001; tic; theta=minFunc(@logistic_regression_vec, theta, options, train.X, train.y); fprintf('Optimization took %f seconds.\n', toc); % Print out training accuracy. tic; accuracy = binary_classifier_accuracy(theta,train.X,train.y); fprintf('Training accuracy: %2.1f%%\n', 100*accuracy); % Print out accuracy on the test set. accuracy = binary_classifier_accuracy(theta,test.X,test.y); fprintf('Test accuracy: %2.1f%%\n', 100*accuracy);
logistic_regression.m
function [f,g] = logistic_regression(theta, X,y) % % Arguments: % theta - A column vector containing the parameter values to optimize. % X - The examples stored in a matrix. % X(i,j) is the i'th coordinate of the j'th example. % y - The label for each example. y(j) is the j'th example's label. % m=size(X,2); n=size(X,1); % initialize objective value and gradient. f = 0; g = zeros(size(theta)); % % TODO: Compute the objective function by looping over the dataset and summing % up the objective values for each example. Store the result in 'f'. % % TODO: Compute the gradient of the objective by looping over the dataset and summing % up the gradients (df/dtheta) for each example. Store the result in 'g'. % %%% YOUR CODE HERE %%% % Step 1?Compute Cost Function for i = 1:m f = f - (y(i)*log(sigmoid(theta' * X(:,i))) + (1-y(i))*log(1-... sigmoid(theta' * X(:,1)))); end for j = 1:n for i = 1:m g(j) = g(j) + X(j,i)*(sigmoid(theta' * X(:,i)) - y(i)); end end
ex1_load_mnist.m (无需更改)
function [train, test] = ex1_load_mnist(binary_digits) % Load the training data X=loadMNISTImages('train-images-idx3-ubyte'); % 784x60000 60000张图片28x28pixel y=loadMNISTLabels('train-labels-idx1-ubyte')'; % 1*60000 if (binary_digits) % Take only the 0 and 1 digits X = [ X(:,y==0), X(:,y==1) ]; %通过y==0和y==1直接得到y=0和1的index y = [ y(y==0), y(y==1) ]; end % Randomly shuffle the data I = randperm(length(y)); y=y(I); % labels in range 1 to 10 X=X(:,I); % We standardize the data so that each pixel will have roughly zero mean and unit variance. s=std(X,[],2); %??std??X???
m=mean(X,2); X=bsxfun(@minus, X, m); X=bsxfun(@rdivide, X, s+.1); % 就是计算(x-m)/s 加0.1是为了防止分母为0 % Place these in the training set train.X = X; train.y = y; % Load the testing data X=loadMNISTImages('t10k-images-idx3-ubyte'); y=loadMNISTLabels('t10k-labels-idx1-ubyte')'; if (binary_digits) % Take only the 0 and 1 digits X = [ X(:,y==0), X(:,y==1) ]; y = [ y(y==0), y(y==1) ]; end % Randomly shuffle the data I = randperm(length(y)); y=y(I); % labels in range 1 to 10 X=X(:,I); % Standardize using the same mean and scale as the training data. X=bsxfun(@minus, X, m); X=bsxfun(@rdivide, X, s+.1); % Place these in the testing set test.X=X; test.y=y;
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