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
%

% forward prop
% J = 1/m * sum(-y .* log(sigmoid(X*theta)) - (1-y) .* log(1 - sigmoid(X*theta))) + (lambda/(2*m)) * sum(theta(2:end).^2)


% recode y
Yvec = zeros(num_labels, length(y));
for yi = 1:length(y),
    Yvec(y(yi),yi) = 1;
end;

% forward prop
A1 = [ones(m, 1) X];
z2 = A1*Theta1';
A2 = sigmoid(z2);
A3 = sigmoid([ones(m, 1) A2]*Theta2');

% calc J
% per output unit
for k = 1:num_labels,
  J += -Yvec(k,:) * log(A3(:,k)) - (1-Yvec(k,:)) * log(1 - A3(:,k));
end;

% TODO: vectorized version


% J unreg
J = J/m;

% J reg
J += (lambda/(2*m)) * ( sum(sum(Theta1(:,2:end) .^ 2)) + sum(sum(Theta2(:,2:end) .^ 2)) );


% 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.
%

Delta1 = Delta2 = 0;

for t = 1:m,
    a1 = [1; X(t,:)']; % activations for layer1
    z2 = Theta1 * a1;
    a2 = [1; sigmoid(z2)];
    z3 = Theta2 * a2;
    a3 = sigmoid(z3);

    % set d3 terms
    d3 = a3 - Yvec(:,t);
    d2 = (Theta2(:,2:end)' * d3) .* sigmoidGradient(z2)
    Delta2 += (d3 * a2');
    Delta1 += (d2 * a1');
end;

Theta1_grad = Delta1/m;
Theta2_grad = Delta2/m;


% 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.
%

Theta1_grad(:,2:end) += (lambda/m) * Theta1(:,2:end);
Theta2_grad(:,2:end) += (lambda/m) * Theta2(:,2:end);


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

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

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


end