costFunctionReg.m

function [J, grad] = costFunctionReg(theta, X, y, lambda)
%COSTFUNCTIONREG Compute cost and gradient for logistic regression with regularization
%   J = COSTFUNCTIONREG(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

% features
n = size(X,2);

h = sigmoid(X*theta);

% Cost function (apenas seguindo a formula)
J = ( (-y)' *log(h)-(1-y)' * log(1-h))/m;

% excluindo o theta0 - devemos ignorar o Theta0 na regularizacao
theta1 = [0 ; theta(2:size(theta), :)];

% somatorio do lambda - REGULARIZATION
soma = 0;
for j=1:n,
	soma = soma + ( theta1(j) * theta1(j) );
end;

% penalty pra cada feature
p = (lambda*soma) / (2*m);

% J + penalty
J = J + p;


% grad.. a formula eh (1/m SOMATORIO ( (h0(x) - y) * X + lambda ).. para j >= 1
% para j == 0 nao devemos considerar a soma com lambda 
% (por isso zeramos a primera posicao do theta1)
grad =  ( X' * (h - y) + lambda*theta1 ) / m;


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

end