1.One-vs-all logistic regression and neural networks to recognize hand-written digits.
Don't just copy paste the code for the sake of completion.
Make sure you understand the code first.
In this exercise, you will implement one-vs-all logistic regression and neural networks to recognize hand-written digits. Before starting the programming exercise, we strongly recommend watching the video lectures and completing the review questions for the associated topics.
It consist of the following files:
- ex3.m - Octave/MATLAB script that steps you through part 1
- ex3 nn.m - Octave/MATLAB script that steps you through part 2
- ex3data1.mat - Training set of hand-written digits
- ex3weights.mat - Initial weights for the neural network exercise
- submit.m - Submission script that sends your solutions to our servers
- displayData.m - Function to help visualize the dataset
- fmincg.m - Function minimization routine (similar to fminunc)
- sigmoid.m - Sigmoid function
- [*] lrCostFunction.m - Logistic regression cost function
- [*] oneVsAll.m - Train a one-vs-all multi-class classifier
- [*] predictOneVsAll.m - Predict using a one-vs-all multi-class classifier
- [*] predict.m - Neural network prediction function
lrCostFunction.m :
function [J, grad] = lrCostFunction(theta, X, y, lambda)
%LRCOSTFUNCTION Compute cost and gradient for logistic regression with
%regularization
% J = LRCOSTFUNCTION(theta, X, y, lambda) computes the cost of using
% theta as the parameter for regularized logistic regression and the
% gradient of the cost w.r.t. to the parameters.
% Initialize some useful values
m = length(y); % number of training examples
% You need to return the following variables correctly
J = 0;
grad = zeros(size(theta));
% ====================== YOUR CODE HERE ======================
% Instructions: Compute the cost of a particular choice of theta.
% You should set J to the cost.
% Compute the partial derivatives and set grad to the partial
% derivatives of the cost w.r.t. each parameter in theta
%
% Hint: The computation of the cost function and gradients can be
% efficiently vectorized. For example, consider the computation
%
% sigmoid(X * theta)
%
% Each row of the resulting matrix will contain the value of the
% prediction for that example. You can make use of this to vectorize
% the cost function and gradient computations.
%
% Hint: When computing the gradient of the regularized cost function,
% there're many possible vectorized solutions, but one solution
% looks like:
% grad = (unregularized gradient for logistic regression)
% temp = theta;
% temp(1) = 0; % because we don't add anything for j = 0
% grad = grad + YOUR_CODE_HERE (using the temp variable)
%
%DIMENSIONS:
% theta = (n+1) x 1
% X = m x (n+1)
% y = m x 1
% grad = (n+1) x 1
% J = Scalar
z = X * theta; % m x 1
h_x = sigmoid(z); % m x 1
reg_term = (lambda/(2*m)) * sum(theta(2:end).^2);
J = (1/m)*sum((-y.*log(h_x))-((1-y).*log(1-h_x))) + reg_term; % scalar
grad(1) = (1/m) * (X(:,1)'*(h_x-y)); % 1 x 1
grad(2:end) = (1/m) * (X(:,2:end)'*(h_x-y)) + (lambda/m)*theta(2:end); % n x 1
% =============================================================
grad = grad(:);
endoneVsAll.m :
function [all_theta] = oneVsAll(X, y, num_labels, lambda)
%ONEVSALL trains multiple logistic regression classifiers and returns all
%the classifiers in a matrix all_theta, where the i-th row of all_theta
%corresponds to the classifier for label i
% [all_theta] = ONEVSALL(X, y, num_labels, lambda) trains num_labels
% logistic regression classifiers and returns each of these classifiers
% in a matrix all_theta, where the i-th row of all_theta corresponds
% to the classifier for label i
% num_labels = No. of output classifier (Here, it is 10)
% Some useful variables
m = size(X, 1); % No. of Training Samples == No. of Images : (Here, 5000)
n = size(X, 2); % No. of features == No. of pixels in each Image : (Here, 400)
% You need to return the following variables correctly
all_theta = zeros(num_labels, n + 1);
%DIMENSIONS: num_labels x (input_layer_size+1) == num_labels x (no_of_features+1) == 10 x 401
%DIMENSIONS: X = m x input_layer_size
%Here, 1 row in X represents 1 training Image of pixel 20x20
% Add ones to the X data matrix
X = [ones(m, 1) X]; %DIMENSIONS: X = m x (input_layer_size+1) = m x (no_of_features+1)
% ====================== YOUR CODE HERE ======================
% Instructions: You should complete the following code to train num_labels
% logistic regression classifiers with regularization
% parameter lambda.
%
% Hint: theta(:) will return a column vector.
%
% Hint: You can use y == c to obtain a vector of 1's and 0's that tell you
% whether the ground truth is true/false for this class.
%
% Note: For this assignment, we recommend using fmincg to optimize the cost
% function. It is okay to use a for-loop (for c = 1:num_labels) to
% loop over the different classes.
%
% fmincg works similarly to fminunc, but is more efficient when we
% are dealing with large number of parameters.
%
% Example Code for fmincg:
%
% % Set Initial theta
% initial_theta = zeros(n + 1, 1);
%
% % Set options for fminunc
% options = optimset('GradObj', 'on', 'MaxIter', 50);
%
% % Run fmincg to obtain the optimal theta
% % This function will return theta and the cost
% [theta] = ...
% fmincg (@(t)(lrCostFunction(t, X, (y == c), lambda)), ...
% initial_theta, options);
%
initial_theta = zeros(n+1, 1);
options = optimset('GradObj', 'on', 'MaxIter', 50);
for c=1:num_labels
all_theta(c,:) = ...
fmincg (@(t)(lrCostFunction(t, X, (y == c), lambda)), ...
initial_theta, options);
end
% =========================================================================
endpredictOneVsAll.m :
function p = predictOneVsAll(all_theta, X)
%PREDICT Predict the label for a trained one-vs-all classifier. The labels
%are in the range 1..K, where K = size(all_theta, 1).
% p = PREDICTONEVSALL(all_theta, X) will return a vector of predictions
% for each example in the matrix X. Note that X contains the examples in
% rows. all_theta is a matrix where the i-th row is a trained logistic
% regression theta vector for the i-th class. You should set p to a vector
% of values from 1..K (e.g., p = [1; 3; 1; 2] predicts classes 1, 3, 1, 2
% for 4 examples)
m = size(X, 1); % No. of Input Examples to Predict (Each row = 1 Example)
num_labels = size(all_theta, 1); %No. of Ouput Classifier
% You need to return the following variables correctly
p = zeros(size(X, 1), 1); % No_of_Input_Examples x 1 == m x 1
% Add ones to the X data matrix
X = [ones(m, 1) X];
% ====================== YOUR CODE HERE ======================
% Instructions: Complete the following code to make predictions using
% your learned logistic regression parameters (one-vs-all).
% You should set p to a vector of predictions (from 1 to
% num_labels).
%
% Hint: This code can be done all vectorized using the max function.
% In particular, the max function can also return the index of the
% max element, for more information see 'help max'. If your examples
% are in rows, then, you can use max(A, [], 2) to obtain the max
% for each row.
%
% num_labels = No. of output classifier (Here, it is 10)
% DIMENSIONS:
% all_theta = 10 x 401 = num_labels x (input_layer_size+1) == num_labels x (no_of_features+1)
prob_mat = X * all_theta'; % 5000 x 10 == no_of_input_image x num_labels
[prob, p] = max(prob_mat,[],2); % m x 1
%returns maximum element in each row == max. probability and its index for each input image
%p: predicted output (index)
%prob: probability of predicted output
%%%%%%%% WORKING: Computation per input image %%%%%%%%%
% for i = 1:m % To iterate through each input sample
% one_image = X(i,:); % 1 x 401 == 1 x no_of_features
% prob_mat = one_image * all_theta'; % 1 x 10 == 1 x num_labels
% [prob, out] = max(prob_mat);
% %out: predicted output
% %prob: probability of predicted output
% p(i) = out;
% end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%% WORKING %%%%%%%%%
% for i = 1:m
% RX = repmat(X(i,:),num_labels,1);
% RX = RX .* all_theta;
% SX = sum(RX,2);
% [val, index] = max(SX);
% p(i) = index;
% end
%%%%%%%%%%%%%%%%%%%%%%%%%%
% =========================================================================
endpredict.m :
function p = predict(Theta1, Theta2, X)
%PREDICT Predict the label of an input given a trained neural network
% p = PREDICT(Theta1, Theta2, X) outputs the predicted label of X given the
% trained weights of a neural network (Theta1, Theta2)
% Useful values
m = size(X, 1);
num_labels = size(Theta2, 1);
% You need to return the following variables correctly
p = zeros(size(X, 1), 1); % m x 1
% ====================== YOUR CODE HERE ======================
% Instructions: Complete the following code to make predictions using
% your learned neural network. You should set p to a
% vector containing labels between 1 to num_labels.
%
% Hint: The max function might come in useful. In particular, the max
% function can also return the index of the max element, for more
% information see 'help max'. If your examples are in rows, then, you
% can use max(A, [], 2) to obtain the max for each row.
%
%DIMENSIONS:
% theta1 = 25 x 401
% theta2 = 10 x 26
% layer1 (input) = 400 nodes + 1bias
% layer2 (hidden) = 25 nodes + 1bias
% layer3 (output) = 10 nodes
%
% theta dimensions = S_(j+1) x ((S_j)+1)
% theta1 = 25 x 401
% theta2 = 10 x 26
% theta1:
% 1st row indicates: theta corresponding to all nodes from layer1 connecting to for 1st node of layer2
% 2nd row indicates: theta corresponding to all nodes from layer1 connecting to for 2nd node of layer2
% and
% 1st Column indicates: theta corresponding to node1 from layer1 to all nodes in layer2
% 2nd Column indicates: theta corresponding to node2 from layer1 to all nodes in layer2
%
% theta2:
% 1st row indicates: theta corresponding to all nodes from layer2 connecting to for 1st node of layer3
% 2nd row indicates: theta corresponding to all nodes from layer2 connecting to for 2nd node of layer3
% and
% 1st Column indicates: theta corresponding to node1 from layer2 to all nodes in layer3
% 2nd Column indicates: theta corresponding to node2 from layer2 to all nodes in layer3
a1 = [ones(m,1) X]; % 5000 x 401 == no_of_input_images x no_of_features % Adding 1 in X
%No. of rows = no. of input images
%No. of Column = No. of features in each image
z2 = a1 * Theta1'; % 5000 x 25
a2 = sigmoid(z2); % 5000 x 25
a2 = [ones(size(a2,1),1) a2]; % 5000 x 26
z3 = a2 * Theta2'; % 5000 x 10
a3 = sigmoid(z3); % 5000 x 10
[prob, p] = max(a3,[],2);
%returns maximum element in each row == max. probability and its index for each input image
%p: predicted output (index)
%prob: probability of predicted output
% =========================================================================
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 [Assignment Solution] - Andrew NG){kind=link}
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