nnCostFunction.m

function [J grad] = nnCostFunction(neuralNetworkParameters, ...
                                   input_layer_size, ...
                                   hidden_layer_size, ...
                                   num_labels, ...
                                   X, y, lambda)
%NNCOSTFUNCTION Implements the neural network cost function for a two layer
%neural network which performs classification
%   [J grad] = NNCOSTFUNCTON(neuralNetworkParameters, hidden_layer_size, num_labels, ...
%   X, y, lambda) computes the cost and gradient of the neural network. The
%   parameters for the neural network are "unrolled" into the vector
%   neuralNetworkParameters and need to be converted back into the weight matrices. 
% 
%   The returned parameter grad should be a "unrolled" vector of the
%   partial derivatives of the neural network.
%

% Reshape neuralNetworkParameters back into the parameters Theta1 and Theta2, the weight matrices
% for our 2 layer neural network
Theta1 = reshape(neuralNetworkParameters(1:hidden_layer_size * (input_layer_size + 1)), ...
                 hidden_layer_size, (input_layer_size + 1));

Theta2 = reshape(neuralNetworkParameters((1 + (hidden_layer_size * (input_layer_size + 1))):end), ...
                 num_labels, (hidden_layer_size + 1));

% Setup some useful variables
numberOfTrainingExamples = size(X, 1);
         
% You need to return the following variables correctly 
J = 0;
Theta1_grad = zeros(size(Theta1));
Theta2_grad = zeros(size(Theta2));

% Part 1: Feedforward the neural network and return the cost in the
%         variable J. After implementing Part 1, you can verify that your
%         cost function computation is correct by verifying the cost
%         computed in the main file's output to console.

% X = 5000 x 400 matrix
X = [ones(numberOfTrainingExamples, 1) X];
% Now X = 5000 x 401 matrix

a1 = X; % a1 = 5000 x 401 matrix

z2 = a1 * Theta1';
a2 = sigmoid(z2);
a2 = [ones(numberOfTrainingExamples, 1) a2];

a3 = sigmoid(a2 * Theta2');

ry = eye(num_labels)(y, :);

cost = ry .* log(a3) + (1 - ry) .* log(1 - a3);
J = -sum( sum(cost, 2) ) / numberOfTrainingExamples;

reg = sum( sum(Theta1(:, 2:end).^2) ) + sum( sum(Theta2(: , 2:end).^2));

J = J + lambda / (2 * numberOfTrainingExamples) * reg;

% Part 2: Implement the backpropagation algorithm to compute the gradients
%         Theta1_grad and Theta2_grad. You should return the partial derivatives of
%         the cost function with respect to Theta1 and Theta2 in Theta1_grad and
%         Theta2_grad, respectively. After implementing Part 2, you can check
%         that your implementation is correct by running checkNNGradients
%
%         Note: The vector y passed into the function is a vector of labels
%               containing values from 1..K. You need to map this vector into a 
%               binary vector of 1's and 0's to be used with the neural network
%               cost function.
%
%         Hint: We recommend implementing backpropagation using a for-loop
%               over the training examples if you are implementing it for the 
%               first time.

% Part 3: Implement regularization with the cost function and gradients.
%
%         Hint: You can implement this around the code for
%               backpropagation. That is, you can compute the gradients for
%               the regularization separately and then add them to Theta1_grad
%               and Theta2_grad from Part 2.

G1 = zeros( size(Theta1) );
G2 = zeros( size(Theta2) );

for i = 1:numberOfTrainingExamples,
	ra1 = X(i, :)';

	rz2 = Theta1 * ra1;
	ra2 = sigmoid(rz2);
	ra2 = [1; ra2];

	rz3 = Theta2 * ra2;
	ra3 = sigmoid(rz3);

	err3 = ra3 - ry(i, :)';

	err2 = (Theta2' * err3)(2:end, 1) .* sigmoidGradient(rz2);

	G1 = G1 + err2 * ra1';
	G2 = G2 + err3 * ra2';
end

Theta1_grad = G1 / numberOfTrainingExamples + lambda * [zeros(hidden_layer_size, 1) Theta1(:, 2:end)] / numberOfTrainingExamples;
Theta2_grad = G2 / numberOfTrainingExamples + lambda * [zeros(num_labels, 1) Theta2(:, 2:end)] / numberOfTrainingExamples;

% Unroll gradients
grad = [Theta1_grad(:) ; Theta2_grad(:)];

end