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);
num_labels = size(Theta2, 1);
X = [ones(m, 1) X];
a1 = X;

z2 = a1*Theta1';
a2 = sigmoid(z2);
a2 = [ones(m, 1) a2];
z3 = a2*Theta2';
a3 = sigmoid(z3);
h = a3;
I = eye(num_labels);
for i = 1:size(y,1)
   Y(i,:) = I(y(i),:); 
end

J = 0;
Theta1_grad = zeros(size(Theta1));
Theta2_grad = zeros(size(Theta2));
l1 = log(h);
l2 = log(1-h);
for i = 1:m
   acu = sum((-Y(i,:)*l1(i,:)' - ((1-Y(i,:))*l2(i,:)')));
   J = J+acu;
   delta_3(i,:) = a3(i,:)-Y(i,:);
   delta_2(i,:) = delta_3(i,:)*Theta2(:,2:end).*sigmoidGradient(z2(i,:));
end
Theta1_grad(:,1) = (Theta1_grad(:,1) + delta_2'*a1(:,1))/m;
Theta2_grad(:,1) = (Theta2_grad(:,1) + delta_3'*a2(:,1))/m;
Theta1_grad(:,2:end) = (Theta1_grad(:,2:end) + delta_2'*a1(:,2:end) + lambda*Theta1(:,2:end))/m;
Theta2_grad(:,2:end) = (Theta2_grad(:,2:end) + delta_3'*a2(:,2:end) +lambda*Theta2(:,2:end))/m;
J = (1/m)*J;
t1 = Theta1(:,2:end);
t1 = t1(:);
t2 = Theta2(:,2:end);
t2 = t2(:);
s = lambda/(2*m)*(t1'*t1+t2'*t2);
J = s+J;

% ====================== 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.
%
% 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.
%

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

end