CheeseZH: Stanford University: Machine Learning Ex3: Multiclass Logistic Regression and Neural Network Prediction

Handwritten digits recognition (0-9)

Multi-class Logistic Regression

1. Vectorizing Logistic Regression

(1) Vectorizing the cost function

(2) Vectorizing the gradient

(3) Vectorizing the regularized cost function

(4) Vectorizing the regularized gradient

All above 4 formulas can be found in the previous blog: click here.

lrCostFunction.m

function [J, grad] = lrCostFunction(theta, X, y, lambda)
%LRCOSTFUNCTION Compute cost and gradient for logistic regression with 
%regularization
%   J = LRCOSTFUNCTION(theta, X, y, lambda) computes the cost of using
%   theta as the parameter for regularized logistic regression and the
%   gradient of the cost w.r.t. to the parameters. 

% Initialize some useful values
m = length(y); % number of training examples

% You need to return the following variables correctly 
J = 0;
grad = zeros(size(theta));

% ====================== YOUR CODE HERE ======================
% Instructions: Compute the cost of a particular choice of theta.
%               You should set J to the cost.
%               Compute the partial derivatives and set grad to the partial
%               derivatives of the cost w.r.t. each parameter in theta
%
% Hint: The computation of the cost function and gradients can be
%       efficiently vectorized. For example, consider the computation
%
%           sigmoid(X * theta)
%
%       Each row of the resulting matrix will contain the value of the
%       prediction for that example. You can make use of this to vectorize
%       the cost function and gradient computations. 
%
% Hint: When computing the gradient of the regularized cost function, 
%       there're many possible vectorized solutions, but one solution
%       looks like:
%           grad = (unregularized gradient for logistic regression)
%           temp = theta; 
%           temp(1) = 0;   % because we don't add anything for j = 0  
%           grad = grad + YOUR_CODE_HERE (using the temp variable)
%

hx = sigmoid(X*theta);
reg = lambda/(2*m)*sum(theta(2:size(theta),:).^2);
J = -1/m*(y'*log(hx)+(1-y)'*log(1-hx)) + reg;
theta(1) = 0;
grad = 1/m*X'*(hx-y)+lambda/m*theta;






% =============================================================

grad = grad(:);

end

2. One-vs-all Classification (Training)

Return all the classifier parameters in a matrix Θ (a K x N+1 matrix, K is the num_labels and N is the num_features ), where each row of Θ corresponds to the learned logistic regression parameters for one class. You can do this with a 'for'-loop from 1 to K, training each classifier independently.

oneVsAll.m

function [all_theta] = oneVsAll(X, y, num_labels, lambda)
%ONEVSALL trains multiple logistic regression classifiers and returns all
%the classifiers in a matrix all_theta, where the i-th row of all_theta 
%corresponds to the classifier for label i
%   [all_theta] = ONEVSALL(X, y, num_labels, lambda) trains num_labels
%   logisitc regression classifiers and returns each of these classifiers
%   in a matrix all_theta, where the i-th row of all_theta corresponds 
%   to the classifier for label i

% Some useful variables
m = size(X, 1);
n = size(X, 2);

% You need to return the following variables correctly 
all_theta = zeros(num_labels, n + 1);

% Add ones to the X data matrix
X = [ones(m, 1) X];

% ====================== YOUR CODE HERE ======================
% Instructions: You should complete the following code to train num_labels
%               logistic regression classifiers with regularization
%               parameter lambda. 
%
% Hint: theta(:) will return a column vector.
%
% Hint: You can use y == c to obtain a vector of 1's and 0's that tell use 
%       whether the ground truth is true/false for this class.
%
% Note: For this assignment, we recommend using fmincg to optimize the cost
%       function. It is okay to use a for-loop (for c = 1:num_labels) to
%       loop over the different classes.
%
%       fmincg works similarly to fminunc, but is more efficient when we
%       are dealing with large number of parameters.
%
% Example Code for fmincg:
%
%     % Set Initial theta
%     initial_theta = zeros(n + 1, 1);
%     
%     % Set options for fminunc
%     options = optimset('GradObj', 'on', 'MaxIter', 50);
% 
%     % Run fmincg to obtain the optimal theta
%     % This function will return theta and the cost 
%     [theta] = ...
%         fmincg (@(t)(lrCostFunction(t, X, (y == c), lambda)), ...
%                 initial_theta, options);
%

for c=1:num_labels,
  initial_theta = all_theta(c,:)';
  options = optimset('GradObj','on','MaxIter',50);
  theta = fmincg(@(t)(lrCostFunction(t,X,(y==c),lambda)),initial_theta,options);
  all_theta(c,:) = theta';
end;


% =========================================================================


end

3. One-vs-all Classification (Prediction)

predictOneVsAll.m

Neural Network Prediction

Feedword Propagation and Prediction

predict.m

function p = predict(Theta1, Theta2, X)
%PREDICT Predict the label of an input given a trained neural network
%   p = PREDICT(Theta1, Theta2, X) outputs the predicted label of X given the
%   trained weights of a neural network (Theta1, Theta2)

% Useful values
m = size(X, 1);
num_labels = size(Theta2, 1);

% You need to return the following variables correctly 
p = zeros(size(X, 1), 1);

% ====================== YOUR CODE HERE ======================
% Instructions: Complete the following code to make predictions using
%               your learned neural network. You should set p to a 
%               vector containing labels between 1 to num_labels.
%
% Hint: The max function might come in useful. In particular, the max
%       function can also return the index of the max element, for more
%       information see 'help max'. If your examples are in rows, then, you
%       can use max(A, [], 2) to obtain the max for each row.
%
a1 = X; %5000*400
a1 = [ones(size(X,1), 1),X]; %5000*401
a2 = sigmoid(a1*Theta1');%5000*25
a2 = [ones(size(a2,1),1),a2]; %5000*26
a3 = sigmoid(a2*Theta2');%5000*10
[tmp,p] = max(a3,[],2);
% =========================================================================


end

Other files and dataset can be download in Coursera.