Neural Network Learning : Machine Learning

My solutions to Week 5 assignment questions.

Helpful links : https://github.com/jcgillespie/Coursera-Machine-Learning/tree/master/ex4

sigmoidGradient.m (#3):

function g = sigmoidGradient(z)
%SIGMOIDGRADIENT returns the gradient of the sigmoid function
%evaluated at z
%   g = SIGMOIDGRADIENT(z) computes the gradient of the sigmoid function
%   evaluated at z. This should work regardless if z is a matrix or a
%   vector. In particular, if z is a vector or matrix, you should return
%   the gradient for each element.

g = zeros(size(z));

% ====================== YOUR CODE HERE ======================
% Instructions: Compute the gradient of the sigmoid function evaluated at
%               each value of z (z can be a matrix, vector or scalar).

sigmoidZ = zeros(size(z, 1), size(z, 2));
for i=1:size(sigmoidZ, 1)
 for j=1:size(sigmoidZ, 2)
  sigmoidZ(i,j) = sigmoid(z(i,j));
 end
end

oneMinus = 1-sigmoidZ;

for i=1:size(g,1)
 for j=1:size(g,2)
  g(i,j) = sigmoidZ(i,j)*oneMinus(i, j);
 end
end

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

end

nnCostFunction(#1. #2. #4. #5):

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.
%




% Ex1 
X = [ones(m, 1) X];
Y = zeros(size(y, 1), num_labels);

for i=1:size(Y, 1) Y(i, y(i)) = 1; end

a2 = zeros(hidden_layer_size, m+1);
a3 = zeros(num_labels, hidden_layer_size+1);
 

a2 = (Theta1*X');
z2 = a2;
a2 = sigmoid(a2); 

a2 = a2';
a2WithoutOnes = a2;
a2 = [ones(size(a2, 1), 1) a2]; 
a3 = a2*Theta2'; 
a3 = sigmoid(a3); 

predictions = a3;
logPredictions = log(predictions); 
 
tempLeftProd = zeros(size(a3, 1), 1);
tempRightProd = zeros(size(a3, 1), 1);

oneMinusY = 1-Y;
oneMinusPredictions = 1-predictions;

for i=1:size(a3, 1)
 tempLeftProd(i) = logPredictions(i,:)*Y(i,:)';
 tempRightProd(i) = log(oneMinusPredictions(i,:))*(oneMinusY(i,:)');
end

brackets = tempLeftProd+tempRightProd;
sumAllExamples = sum(brackets);
J = (-1/m)*sumAllExamples;





% Ex2
regularizationAdd = 0;
regAddLeft = zeros(hidden_layer_size, 1);

for i=1:hidden_layer_size
 for j=1:size(Theta1, 2)-1
  regAddLeft(i) = regAddLeft(i) + Theta1(i, j+1)^2;
 end
end

regAddRight = zeros(num_labels, 1);
for i=1:num_labels
 for j=1:size(Theta2, 2)-1 
  regAddRight(i) = regAddRight(i) + Theta2(i, j+1)^2;
 end
end

regularizationAdd = (lambda*(sum(regAddLeft)+sum(regAddRight)))/(2*m);
J = J+regularizationAdd; 






% Ex4
Delta1=0;
Delta2=0;

Theta1WithoutBias = Theta1(:, 2:end);
Theta2WithoutBias = Theta2(:, 2:end);

for t=1:m
 a1 = X(t, :)';
 z2 = Theta1*a1;
 a2 = [1; sigmoid(z2)];
 z3 = Theta2*a2;
 a3 = [sigmoid(z3)];
 
 d3 = a3-Y(t, :)'; 
 
 d2 = Theta2WithoutBias'*d3 .* sigmoidGradient(z2);
 %d2 = d2(2:end); % No need to do that. Theta2WithoutBias 
      % and z2(we add bias to a2, not z2) have 
      % already taken care of that
 
 Delta2 = Delta2 + d3*a2';
 Delta1 = Delta1 + d2*a1';
end

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





% Ex5 - regularization
Theta1_grad(:, 2:end) = Theta1_grad(:, 2:end)+(lambda/m)*Theta1WithoutBias;
Theta2_grad(:, 2:end) = Theta2_grad(:, 2:end)+(lambda/m)*Theta2WithoutBias;
% -------------------------------------------------------------

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

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

end