nnCostFunction.m

function [J grad] = nnCostFunction(nn_params, ...
                                   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(nn_params, 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
%   nn_params 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 nn_params back into the parameters Theta1 and Theta2, the weight matrices
% for our 2 layer neural network
Theta1 = reshape(nn_params(1:hidden_layer_size * (input_layer_size + 1)), ...
                 hidden_layer_size, (input_layer_size + 1));

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

% Setup some useful variables
m = size(X, 1);

% You need to return the following variables correctly
J = 0;
Theta1_grad = zeros(size(Theta1));
Theta2_grad = zeros(size(Theta2));

% ====================== YOUR CODE HERE ======================
% Instructions: You should complete the code by working through the
%               following parts.
%
% 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 ex4.m
%
% 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.
%

a_1 = [ones(1, m); X'];

a_2 = [ones(1, m); sigmoid(Theta1 * a_1)];

hypothesis = sigmoid(Theta2 * a_2);

y_as_vectors = repmat(y', num_labels, 1) == repmat([1:num_labels]', 1, m);

unregJ = (1/m) * sum(sum(((-1 .* y_as_vectors) .* log(hypothesis)) - ((1 .- y_as_vectors) .* log(1 - hypothesis))));

Theta1_to_regularize = [zeros(size(Theta1, 1), 1) Theta1(:, 2:end)];
Theta2_to_regularize = [zeros(size(Theta2, 1), 1) Theta2(:, 2:end)];
all_theta_to_regularize = [Theta1_to_regularize(:) ; Theta2_to_regularize(:)]';
% add regularization:
J = unregJ + ((lambda/(2*m)) * sum(all_theta_to_regularize .^ 2));


% gradients:

for t = 1:m
    a_1_t = a_1(:, t);
    a_2_t = a_2(:, t);

    hypothesis_t = hypothesis(:, t);
    y_t = y_as_vectors(:, t);

    d_3 = hypothesis_t - y_t;
    d_2 = (Theta2' * d_3) .* sigmoidGradient(a_2_t);

    Theta2_grad = Theta2_grad + (d_3*a_2_t');
    Theta1_grad = Theta1_grad + (d_2(2:end)*a_1_t');
endfor

Theta2_grad = (1/m) .* Theta2_grad;
Theta1_grad = (1/m) .* Theta1_grad;


% d_3 = (hypothesis - y_as_vectors);
% d_2 = (Theta2' * d_3) .* sigmoidGradient(a_2);

% Theta2_grad = (1/m) * (d_3 * a_2');
% Theta1_grad = (1/m) * (d_2 * a_1')(2:end, :);

% -------------------------------------------------------------

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

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


end