# Coursera: Machine Learning (Week 5) [Assignment Solution] - Andrew NG

▸ Back-propagation algorithm for neural networks to the task of hand-written digit recognition.

I have recently completed the Machine Learning course from Coursera by Andrew NG.

While doing the course we have to go through various quiz and assignments.

Here, I am sharing my solutions for the weekly assignments throughout the course.

In this exercise, you will implement the back-propagation algorithm for neural networks and apply it to the task of hand-written digit recognition. Before starting on the programming exercise, we strongly recommend watching the video lectures and completing the review questions for the associated topics.

I tried to provide optimized solutions like

I have recently completed the Machine Learning course from Coursera by Andrew NG.

While doing the course we have to go through various quiz and assignments.

Here, I am sharing my solutions for the weekly assignments throughout the course.

**These solutions are for reference only.****> It is recommended that you should solve the assignments by yourself honestly then only it makes sense to complete the course.****>**

**But, In case you stuck in between, feel free to refer to the solutions provided by me.**

**NOTE:**

Don't just copy paste the code for the sake of completion.

Even if you copy the code, make sure you understand the code first.

**Click here to check out**__week-4__assignment solutions,__Scroll down__for the solutions for__week-5__assignment.In this exercise, you will implement the back-propagation algorithm for neural networks and apply it to the task of hand-written digit recognition. Before starting on 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:****ex4.m -**Octave/MATLAB script that steps you through the exercise**ex4data1.mat -**Training set of hand-written digits**ex4weights.mat -**Neural network parameters for exercise 4**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**computeNumericalGradient.m -**Numerically compute gradients**checkNNGradients.m -**Function to help check your gradients**debugInitializeWeights.m -**Function for initializing weights**predict.m -**Neural network prediction function**[*] sigmoidGradient.m -**Compute the gradient of the sigmoid function**[*] randInitializeWeights.m -**Randomly initialize weights**[*] nnCostFunction.m -**Neural network cost function**Video -**YouTube videos featuring Free IOT/ML tutorials

*****indicates files you will need to complete**sigmoidGradient.m :**

```
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).
g = sigmoid(z).*(1-sigmoid(z));
% =============================================================
end
```

**randInitializeWeights.m :**

```
function W = randInitializeWeights(L_in, L_out)
%RANDINITIALIZEWEIGHTS Randomly initialize the weights of a layer with L_in
%incoming connections and L_out outgoing connections
% W = RANDINITIALIZEWEIGHTS(L_in, L_out) randomly initializes the weights
% of a layer with L_in incoming connections and L_out outgoing
% connections.
%
% Note that W should be set to a matrix of size(L_out, 1 + L_in) as
% the first column of W handles the "bias" terms
%
% You need to return the following variables correctly
W = zeros(L_out, 1 + L_in);
% ====================== YOUR CODE HERE ======================
% Instructions: Initialize W randomly so that we break the symmetry while
% training the neural network.
%
% Note: The first column of W corresponds to the parameters for the bias unit
%
% epsilon_init = 0.12;
epsilon_init = sqrt(6)/(sqrt(L_in)+sqrt(L_out));
W = - epsilon_init + rand(L_out, 1 + L_in) * 2 * epsilon_init ;
% =========================================================================
end
```

**Check-out our free tutorials on IOT (Internet of Things):**

**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
% DIMENSIONS:
% Theta1 = 25 x 401
% Theta2 = 10 x 26
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)); %25 x401
Theta2_grad = zeros(size(Theta2)); %10 x 26
% ====================== 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.
%
%%%%%%%%%%% Part 1: Calculating J w/o Regularization %%%%%%%%%%%%%%%
X = [ones(m,1), X]; % Adding 1 as first column in X
a1 = X; % 5000 x 401
z2 = a1 * Theta1'; % m x hidden_layer_size == 5000 x 25
a2 = sigmoid(z2); % m x hidden_layer_size == 5000 x 25
a2 = [ones(size(a2,1),1), a2]; % Adding 1 as first column in z = (Adding bias unit) % m x (hidden_layer_size + 1) == 5000 x 26
z3 = a2 * Theta2'; % m x num_labels == 5000 x 10
a3 = sigmoid(z3); % m x num_labels == 5000 x 10
h_x = a3; % m x num_labels == 5000 x 10
%Converting y into vector of 0's and 1's for multi-class classification
%%%%% WORKING %%%%%
% y_Vec = zeros(m,num_labels);
% for i = 1:m
% y_Vec(i,y(i)) = 1;
% end
%%%%%%%%%%%%%%%%%%%
y_Vec = (1:num_labels)==y; % m x num_labels == 5000 x 10
%Costfunction Without regularization
J = (1/m) * sum(sum((-y_Vec.*log(h_x))-((1-y_Vec).*log(1-h_x)))); %scalar
%%%%%%%%%%% Part 2: Implementing Backpropogation for Theta_gra w/o Regularization %%%%%%%%%%%%%
%%%%%%% WORKING: Backpropogation using for loop %%%%%%%
% for t=1:m
% % Here X is including 1 column at begining
%
% % for layer-1
% a1 = X(t,:)'; % (n+1) x 1 == 401 x 1
%
% % for layer-2
% z2 = Theta1 * a1; % hidden_layer_size x 1 == 25 x 1
% a2 = [1; sigmoid(z2)]; % (hidden_layer_size+1) x 1 == 26 x 1
%
% % for layer-3
% z3 = Theta2 * a2; % num_labels x 1 == 10 x 1
% a3 = sigmoid(z3); % num_labels x 1 == 10 x 1
%
% yVector = (1:num_labels)'==y(t); % num_labels x 1 == 10 x 1
%
% %calculating delta values
% delta3 = a3 - yVector; % num_labels x 1 == 10 x 1
%
% delta2 = (Theta2' * delta3) .* [1; sigmoidGradient(z2)]; % (hidden_layer_size+1) x 1 == 26 x 1
%
% delta2 = delta2(2:end); % hidden_layer_size x 1 == 25 x 1 %Removing delta2 for bias node
%
% % delta_1 is not calculated because we do not associate error with the input
%
% % CAPITAL delta update
% Theta1_grad = Theta1_grad + (delta2 * a1'); % 25 x 401
% Theta2_grad = Theta2_grad + (delta3 * a2'); % 10 x 26
%
% end
%
% Theta1_grad = (1/m) * Theta1_grad; % 25 x 401
% Theta2_grad = (1/m) * Theta2_grad; % 10 x 26
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%% WORKING: Backpropogation (Vectorized Implementation) %%%%%%%
% Here X is including 1 column at begining
A1 = X; % 5000 x 401
Z2 = A1 * Theta1'; % m x hidden_layer_size == 5000 x 25
A2 = sigmoid(Z2); % m x hidden_layer_size == 5000 x 25
A2 = [ones(size(A2,1),1), A2]; % Adding 1 as first column in z = (Adding bias unit) % m x (hidden_layer_size + 1) == 5000 x 26
Z3 = A2 * Theta2'; % m x num_labels == 5000 x 10
A3 = sigmoid(Z3); % m x num_labels == 5000 x 10
% h_x = a3; % m x num_labels == 5000 x 10
y_Vec = (1:num_labels)==y; % m x num_labels == 5000 x 10
DELTA3 = A3 - y_Vec; % 5000 x 10
DELTA2 = (DELTA3 * Theta2) .* [ones(size(Z2,1),1) sigmoidGradient(Z2)]; % 5000 x 26
DELTA2 = DELTA2(:,2:end); % 5000 x 25 %Removing delta2 for bias node
Theta1_grad = (1/m) * (DELTA2' * A1); % 25 x 401
Theta2_grad = (1/m) * (DELTA3' * A2); % 10 x 26
%%%%%%%%%%%% WORKING: DIRECT CALCULATION OF THETA GRADIENT WITH REGULARISATION %%%%%%%%%%%
% %Regularization term is later added in Part 3
% Theta1_grad = (1/m) * Theta1_grad + (lambda/m) * [zeros(size(Theta1, 1), 1) Theta1(:,2:end)]; % 25 x 401
% Theta2_grad = (1/m) * Theta2_grad + (lambda/m) * [zeros(size(Theta2, 1), 1) Theta2(:,2:end)]; % 10 x 26
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%% Part 3: Adding Regularisation term in J and Theta_grad %%%%%%%%%%%%%
reg_term = (lambda/(2*m)) * (sum(sum(Theta1(:,2:end).^2)) + sum(sum(Theta2(:,2:end).^2))); %scalar
%Costfunction With regularization
J = J + reg_term; %scalar
%Calculating gradients for the regularization
Theta1_grad_reg_term = (lambda/m) * [zeros(size(Theta1, 1), 1) Theta1(:,2:end)]; % 25 x 401
Theta2_grad_reg_term = (lambda/m) * [zeros(size(Theta2, 1), 1) Theta2(:,2:end)]; % 10 x 26
%Adding regularization term to earlier calculated Theta_grad
Theta1_grad = Theta1_grad + Theta1_grad_reg_term;
Theta2_grad = Theta2_grad + Theta2_grad_reg_term;
% -------------------------------------------------------------
% =========================================================================
% Unroll gradients
grad = [Theta1_grad(:) ; Theta2_grad(:)];
end
```

I tried to provide optimized solutions like

**vectorized implementation**for each assignment. If you think that more optimization can be done, then put suggest the corrections / improvements.--------------------------------------------------------------------------------

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Feel free to ask doubts in the comment section. I will try my best to solve it.

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Thanks and Regards,

**-Akshay P. Daga**

Hi,

ReplyDeleteI am clear up to how to calculate partial derivatives. But, I am having doubt after calculating delta values. I have got delta-2 values in the dimension 10 X 25 and delta-1 with dimension 25X400. This is I have got for first row of input layer. So, for 5000 rows how these delta values will be calculated?

Thanks.