# 04 Octave Programming: Neural Network Learning

[ 04 Octave Programming: Neural Network Learning ]

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Lecture Note

[ Modelling ]

[ Files ]

ex4.m

```%% Machine Learning Online Class - Exercise 4 Neural Network Learning

%  Instructions
%  ------------
%
%  This file contains code that helps you get started on the
%  linear exercise. You will need to complete the following functions
%  in this exericse:
%
%     randInitializeWeights.m
%     nnCostFunction.m
%
%  For this exercise, you will not need to change any code in this file,
%  or any other files other than those mentioned above.
%

%% Initialization
clear ; close all; clc

%% Setup the parameters you will use for this exercise
input_layer_size  = 400;  % 20x20 Input Images of Digits
hidden_layer_size = 25;   % 25 hidden units
num_labels = 10;          % 10 labels, from 1 to 10
% (note that we have mapped &quot;0&quot; to label 10)

%  We start the exercise by first loading and visualizing the dataset.
%  You will be working with a dataset that contains handwritten digits.
%

m = size(X, 1);

% Randomly select 100 data points to display
sel = randperm(size(X, 1));
sel = sel(1:100);

displayData(X(sel, :));

fprintf('Program paused. Press enter to continue.\n');
pause;

% In this part of the exercise, we load some pre-initialized
% neural network parameters.

% Load the weights into variables Theta1 and Theta2

% Unroll parameters
nn_params = [Theta1(:) ; Theta2(:)];

%% ================ Part 3: Compute Cost (Feedforward) ================
%  To the neural network, you should first start by implementing the
%  feedforward part of the neural network that returns the cost only. You
%  should complete the code in nnCostFunction.m to return cost. After
%  implementing the feedforward to compute the cost, you can verify that
%  your implementation is correct by verifying that you get the same cost
%  as us for the fixed debugging parameters.
%
%  We suggest implementing the feedforward cost *without* regularization
%  first so that it will be easier for you to debug. Later, in part 4, you
%  will get to implement the regularized cost.
%
fprintf('\nFeedforward Using Neural Network ...\n')

% Weight regularization parameter (we set this to 0 here).
lambda = 0;

J = nnCostFunction(nn_params, input_layer_size, hidden_layer_size, ...
num_labels, X, y, lambda);

fprintf(['Cost at parameters (loaded from ex4weights): %f '...
'\n(this value should be about 0.287629)\n'], J);

fprintf('\nProgram paused. Press enter to continue.\n');
pause;

%% =============== Part 4: Implement Regularization ===============
%  Once your cost function implementation is correct, you should now
%  continue to implement the regularization with the cost.
%

fprintf('\nChecking Cost Function (w/ Regularization) ... \n')

% Weight regularization parameter (we set this to 1 here).
lambda = 1;

J = nnCostFunction(nn_params, input_layer_size, hidden_layer_size, ...
num_labels, X, y, lambda);

fprintf(['Cost at parameters (loaded from ex4weights): %f '...
'\n(this value should be about 0.383770)\n'], J);

fprintf('Program paused. Press enter to continue.\n');
pause;

%% ================ Part 5: Sigmoid Gradient  ================
%  Before you start implementing the neural network, you will first
%  implement the gradient for the sigmoid function. You should complete the
%  code in the sigmoidGradient.m file.
%

g = sigmoidGradient([1 -0.5 0 0.5 1]);
fprintf('Sigmoid gradient evaluated at [1 -0.5 0 0.5 1]:\n  ');
fprintf('%f ', g);
fprintf('\n\n');

fprintf('Program paused. Press enter to continue.\n');
pause;

%% ================ Part 6: Initializing Pameters ================
%  In this part of the exercise, you will be starting to implment a two
%  layer neural network that classifies digits. You will start by
%  implementing a function to initialize the weights of the neural network
%  (randInitializeWeights.m)

fprintf('\nInitializing Neural Network Parameters ...\n')

initial_Theta1 = randInitializeWeights(input_layer_size, hidden_layer_size);
initial_Theta2 = randInitializeWeights(hidden_layer_size, num_labels);

% Unroll parameters
initial_nn_params = [initial_Theta1(:) ; initial_Theta2(:)];

%% =============== Part 7: Implement Backpropagation ===============
%  Once your cost matches up with ours, you should proceed to implement the
%  backpropagation algorithm for the neural network. You should add to the
%  code you've written in nnCostFunction.m to return the partial
%  derivatives of the parameters.
%
fprintf('\nChecking Backpropagation... \n');

fprintf('\nProgram paused. Press enter to continue.\n');
pause;

%% =============== Part 8: Implement Regularization ===============
%  Once your backpropagation implementation is correct, you should now
%  continue to implement the regularization with the cost and gradient.
%

fprintf('\nChecking Backpropagation (w/ Regularization) ... \n')

lambda = 3;

% Also output the costFunction debugging values
debug_J  = nnCostFunction(nn_params, input_layer_size, ...
hidden_layer_size, num_labels, X, y, lambda);

fprintf(['\n\nCost at (fixed) debugging parameters (w/ lambda = 10): %f ' ...
'\n(this value should be about 0.576051)\n\n'], debug_J);

fprintf('Program paused. Press enter to continue.\n');
pause;

%% =================== Part 8: Training NN ===================
%  You have now implemented all the code necessary to train a neural
%  network. To train your neural network, we will now use &quot;fmincg&quot;, which
%  is a function which works similarly to &quot;fminunc&quot;. Recall that these
%  advanced optimizers are able to train our cost functions efficiently as
%  long as we provide them with the gradient computations.
%
fprintf('\nTraining Neural Network... \n')

%  After you have completed the assignment, change the MaxIter to a larger
%  value to see how more training helps.
options = optimset('MaxIter', 50);

%  You should also try different values of lambda
lambda = 1;

% Create &quot;short hand&quot; for the cost function to be minimized
costFunction = @(p) nnCostFunction(p, ...
input_layer_size, ...
hidden_layer_size, ...
num_labels, X, y, lambda);

% Now, costFunction is a function that takes in only one argument (the
% neural network parameters)
[nn_params, cost] = fmincg(costFunction, initial_nn_params, options);

% Obtain Theta1 and Theta2 back from nn_params
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));

fprintf('Program paused. Press enter to continue.\n');
pause;

%% ================= Part 9: Visualize Weights =================
%  You can now &quot;visualize&quot; what the neural network is learning by
%  displaying the hidden units to see what features they are capturing in
%  the data.

fprintf('\nVisualizing Neural Network... \n')

displayData(Theta1(:, 2:end));

fprintf('\nProgram paused. Press enter to continue.\n');
pause;

%% ================= Part 10: Implement Predict =================
%  After training the neural network, we would like to use it to predict
%  the labels. You will now implement the &quot;predict&quot; function to use the
%  neural network to predict the labels of the training set. This lets
%  you compute the training set accuracy.

pred = predict(Theta1, Theta2, X);

fprintf('\nTraining Set Accuracy: %f\n', mean(double(pred == y)) * 100);

```

```function g = sigmoidGradient(z)
%evaluated at z
%   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 column row of W handles the &quot;bias&quot; 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 row of W corresponds to the parameters for the bias units
%

% Randomly initialize the weights to small values
epsilon_init = 0.12;
W = rand(L_out, 1 + L_in) * 2 * epsilon_init - epsilon_init;

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

end
```

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 &quot;unrolled&quot; into the vector
%   nn_params and need to be converted back into the weight matrices.
%
%   The returned parameter grad should be a &quot;unrolled&quot; 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;

% ====================== 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
%         the cost function with respect to Theta1 and Theta2 in Theta1_grad and
%         Theta2_grad, respectively. After implementing Part 2, you can check
%
%         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
%               and Theta2_grad from Part 2.
%

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

z2 = a1*Theta1';
a2 = sigmoid(z2);
a2 = [ones(m, 1) a2];

z3 = a2*Theta2';
a3 = sigmoid(z3);
h = a3;

Y = zeros(m,num_labels);
dis = zeros(m,1);

for i = 1:m
Y(i,y(i)) = 1;
dis(i) = log(h(i,:))*(-Y(i,:))' - log(1-h(i,:))*(1-Y(i,:))';
end
J = 1/m * sum(dis);

%add the cost for the regularization terms
J = J + lambda/(2*m)*( sum(sum(Theta1(:,2:end).^2)) + sum(sum(Theta2(:,2:end).^2)) );

%Backpropagation
for t = 1:m
a1 = [1; X(t,:)'];

z2 = Theta1 * a1;
a2 = [1;sigmoid(z2)];

z3 = Theta2 * a2;
a3 = sigmoid(z3);

delta3 = a3 - Y(t,:)';

delta2 = delta2(2:end);

end
%regularization

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

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

end
```

displayData.m

```function [h, display_array] = displayData(X, example_width)
%DISPLAYDATA Display 2D data in a nice grid
%   [h, display_array] = DISPLAYDATA(X, example_width) displays 2D data
%   stored in X in a nice grid. It returns the figure handle h and the
%   displayed array if requested.

% Set example_width automatically if not passed in
if ~exist('example_width', 'var') || isempty(example_width)
example_width = round(sqrt(size(X, 2)));
end

% Gray Image
colormap(gray);

% Compute rows, cols
[m n] = size(X);
example_height = (n / example_width);

% Compute number of items to display
display_rows = floor(sqrt(m));
display_cols = ceil(m / display_rows);

% Setup blank display

% Copy each example into a patch on the display array
curr_ex = 1;
for j = 1:display_rows
for i = 1:display_cols
if curr_ex &gt; m,
break;
end
% Copy the patch

% Get the max value of the patch
max_val = max(abs(X(curr_ex, :)));
display_array(pad + (j - 1) * (example_height + pad) + (1:example_height), ...
pad + (i - 1) * (example_width + pad) + (1:example_width)) = ...
reshape(X(curr_ex, :), example_height, example_width) / max_val;
curr_ex = curr_ex + 1;
end
if curr_ex &gt; m,
break;
end
end

% Display Image
h = imagesc(display_array, [-1 1]);

% Do not show axis
axis image off

drawnow;

end
```

```function checkNNGradients(lambda)
%CHECKNNGRADIENTS Creates a small neural network to check the
%   CHECKNNGRADIENTS(lambda) Creates a small neural network to check the
%   result in very similar values.
%

if ~exist('lambda', 'var') || isempty(lambda)
lambda = 0;
end

input_layer_size = 3;
hidden_layer_size = 5;
num_labels = 3;
m = 5;

% We generate some 'random' test data
Theta1 = debugInitializeWeights(hidden_layer_size, input_layer_size);
Theta2 = debugInitializeWeights(num_labels, hidden_layer_size);
% Reusing debugInitializeWeights to generate X
X  = debugInitializeWeights(m, input_layer_size - 1);
y  = 1 + mod(1:m, num_labels)';

% Unroll parameters
nn_params = [Theta1(:) ; Theta2(:)];

% Short hand for cost function
costFunc = @(p) nnCostFunction(p, input_layer_size, hidden_layer_size, ...
num_labels, X, y, lambda);

% Visually examine the two gradient computations.  The two columns
% you get should be very similar.
fprintf(['The above two columns you get should be very similar.\n' ...

% Evaluate the norm of the difference between two solutions.
% If you have a correct implementation, and assuming you used EPSILON = 0.0001
% in computeNumericalGradient.m, then diff below should be less than 1e-9

fprintf(['If your backpropagation implementation is correct, then \n' ...
'the relative difference will be small (less than 1e-9). \n' ...
'\nRelative Difference: %g\n'], diff);

end
```

```function numgrad = computeNumericalGradient(J, theta)
%and gives us a numerical estimate of the gradient.
%   gradient of the function J around theta. Calling y = J(theta) should
%   return the function value at theta.

% Notes: The following code implements numerical gradient checking, and
%        approximation of) the partial derivative of J with respect to the
%        i-th input argument, evaluated at theta. (i.e., numgrad(i) should
%        be the (approximately) the partial derivative of J with respect
%        to theta(i).)
%

perturb = zeros(size(theta));
e = 1e-4;
for p = 1:numel(theta)
% Set perturbation vector
perturb(p) = e;
loss1 = J(theta - perturb);
loss2 = J(theta + perturb);
numgrad(p) = (loss2 - loss1) / (2*e);
perturb(p) = 0;
end

end
```

debugInitializeWeights.m

```function W = debugInitializeWeights(fan_out, fan_in)
%DEBUGINITIALIZEWEIGHTS Initialize the weights of a layer with fan_in
%incoming connections and fan_out outgoing connections using a fixed
%   W = DEBUGINITIALIZEWEIGHTS(fan_in, fan_out) initializes the weights
%   of a layer with fan_in incoming connections and fan_out outgoing
%   connections using a fix set of values
%
%   Note that W should be set to a matrix of size(1 + fan_in, fan_out) as
%   the first row of W handles the &quot;bias&quot; terms
%

% Set W to zeros
W = zeros(fan_out, 1 + fan_in);

% Initialize W using &quot;sin&quot;, this ensures that W is always of the same
% values and will be useful for debugging
W = reshape(sin(1:numel(W)), size(W)) / 10;

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

end
```

fmincg.m

```function [X, fX, i] = fmincg(f, X, options, P1, P2, P3, P4, P5)
% Minimize a continuous differentialble multivariate function. Starting point
% is given by &quot;X&quot; (D by 1), and the function named in the string &quot;f&quot;, must
% return a function value and a vector of partial derivatives. The Polack-
% Ribiere flavour of conjugate gradients is used to compute search directions,
% and a line search using quadratic and cubic polynomial approximations and the
% Wolfe-Powell stopping criteria is used together with the slope ratio method
% for guessing initial step sizes. Additionally a bunch of checks are made to
% make sure that exploration is taking place and that extrapolation will not
% be unboundedly large. The &quot;length&quot; gives the length of the run: if it is
% positive, it gives the maximum number of line searches, if negative its
% absolute gives the maximum allowed number of function evaluations. You can
% (optionally) give &quot;length&quot; a second component, which will indicate the
% reduction in function value to be expected in the first line-search (defaults
% to 1.0). The function returns when either its length is up, or if no further
% progress can be made (ie, we are at a minimum, or so close that due to
% numerical problems, we cannot get any closer). If the function terminates
% within a few iterations, it could be an indication that the function value
% and derivatives are not consistent (ie, there may be a bug in the
% implementation of your &quot;f&quot; function). The function returns the found
% solution &quot;X&quot;, a vector of function values &quot;fX&quot; indicating the progress made
% and &quot;i&quot; the number of iterations (line searches or function evaluations,
% depending on the sign of &quot;length&quot;) used.
%
% Usage: [X, fX, i] = fmincg(f, X, options, P1, P2, P3, P4, P5)
%
%
% Copyright (C) 2001 and 2002 by Carl Edward Rasmussen. Date 2002-02-13
%
%
% (C) Copyright 1999, 2000 &amp; 2001, Carl Edward Rasmussen
%
% Permission is granted for anyone to copy, use, or modify these
% programs and accompanying documents for purposes of research or
% education, provided this copyright notice is retained, and note is
%
% These programs and documents are distributed without any warranty,
% express or implied.  As the programs were written for research
% purposes only, they have not been tested to the degree that would be
% advisable in any important application.  All use of these programs is
% entirely at the user's own risk.
%
% 1) Function name and argument specifications
% 2) Output display
%

if exist('options', 'var') &amp;&amp; ~isempty(options) &amp;&amp; isfield(options, 'MaxIter')
length = options.MaxIter;
else
length = 100;
end

RHO = 0.01;                            % a bunch of constants for line searches
SIG = 0.5;       % RHO and SIG are the constants in the Wolfe-Powell conditions
INT = 0.1;    % don't reevaluate within 0.1 of the limit of the current bracket
EXT = 3.0;                    % extrapolate maximum 3 times the current bracket
MAX = 20;                         % max 20 function evaluations per line search
RATIO = 100;                                      % maximum allowed slope ratio

argstr = ['feval(f, X'];                      % compose string used to call function
for i = 1:(nargin - 3)
argstr = [argstr, ',P', int2str(i)];
end
argstr = [argstr, ')'];

if max(size(length)) == 2, red=length(2); length=length(1); else red=1; end
S=['Iteration '];

i = 0;                                            % zero the run length counter
ls_failed = 0;                             % no previous line search has failed
fX = [];
[f1 df1] = eval(argstr);                      % get function value and gradient
i = i + (length&lt;0);                                            % count epochs?!
s = -df1;                                        % search direction is steepest
d1 = -s'*s;                                                 % this is the slope
z1 = red/(1-d1);                                  % initial step is red/(|s|+1)

while i &lt; abs(length)                                      % while not finished   i = i + (length&gt;0);                                      % count iterations?!

X0 = X; f0 = f1; df0 = df1;                   % make a copy of current values
X = X + z1*s;                                             % begin line search
[f2 df2] = eval(argstr);
i = i + (length&lt;0);                                          % count epochs?!   d2 = df2'*s;   f3 = f1; d3 = d1; z3 = -z1;             % initialize point 3 equal to point 1   if length&gt;0, M = MAX; else M = min(MAX, -length-i); end
success = 0; limit = -1;                     % initialize quanteties
while 1
while ((f2 &gt; f1+z1*RHO*d1) || (d2 &gt; -SIG*d1)) &amp;&amp; (M &gt; 0)
limit = z1;                                         % tighten the bracket
if f2 &gt; f1
z2 = z3 - (0.5*d3*z3*z3)/(d3*z3+f2-f3);                 % quadratic fit
else
A = 6*(f2-f3)/z3+3*(d2+d3);                                 % cubic fit
B = 3*(f3-f2)-z3*(d3+2*d2);
z2 = (sqrt(B*B-A*d2*z3*z3)-B)/A;       % numerical error possible - ok!
end
if isnan(z2) || isinf(z2)
z2 = z3/2;                  % if we had a numerical problem then bisect
end
z2 = max(min(z2, INT*z3),(1-INT)*z3);  % don't accept too close to limits
z1 = z1 + z2;                                           % update the step
X = X + z2*s;
[f2 df2] = eval(argstr);
M = M - 1; i = i + (length&lt;0);                           % count epochs?!       d2 = df2'*s;       z3 = z3-z2;                    % z3 is now relative to the location of z2     end     if f2 &gt; f1+z1*RHO*d1 || d2 &gt; -SIG*d1
break;                                                % this is a failure
elseif d2 &gt; SIG*d1
success = 1; break;                                             % success
elseif M == 0
break;                                                          % failure
end
A = 6*(f2-f3)/z3+3*(d2+d3);                      % make cubic extrapolation
B = 3*(f3-f2)-z3*(d3+2*d2);
z2 = -d2*z3*z3/(B+sqrt(B*B-A*d2*z3*z3));        % num. error possible - ok!
if ~isreal(z2) || isnan(z2) || isinf(z2) || z2 &lt; 0 % num prob or wrong sign?
if limit &lt; -0.5                               % if we have no upper limit         z2 = z1 * (EXT-1);                 % the extrapolate the maximum amount       else         z2 = (limit-z1)/2;                                   % otherwise bisect       end     elseif (limit &gt; -0.5) &amp;&amp; (z2+z1 &gt; limit)         % extraplation beyond max?
z2 = (limit-z1)/2;                                               % bisect
elseif (limit &lt; -0.5) &amp;&amp; (z2+z1 &gt; z1*EXT)       % extrapolation beyond limit
z2 = z1*(EXT-1.0);                           % set to extrapolation limit
elseif z2 &lt; -z3*INT       z2 = -z3*INT;     elseif (limit &gt; -0.5) &amp;&amp; (z2 &lt; (limit-z1)*(1.0-INT))  % too close to limit?
z2 = (limit-z1)*(1.0-INT);
end
f3 = f2; d3 = d2; z3 = -z2;                  % set point 3 equal to point 2
z1 = z1 + z2; X = X + z2*s;                      % update current estimates
[f2 df2] = eval(argstr);
M = M - 1; i = i + (length&lt;0);                             % count epochs?!     d2 = df2'*s;   end                                                      % end of line search   if success                                         % if line search succeeded     f1 = f2; fX = [fX' f1]';     fprintf('%s %4i | Cost: %4.6e\r', S, i, f1);     s = (df2'*df2-df1'*df2)/(df1'*df1)*s - df2;      % Polack-Ribiere direction     tmp = df1; df1 = df2; df2 = tmp;                         % swap derivatives     d2 = df1'*s;     if d2 &gt; 0                                      % new slope must be negative
s = -df1;                              % otherwise use steepest direction
d2 = -s'*s;
end
z1 = z1 * min(RATIO, d1/(d2-realmin));          % slope ratio but max RATIO
d1 = d2;
ls_failed = 0;                              % this line search did not fail
else
X = X0; f1 = f0; df1 = df0;  % restore point from before failed line search
if ls_failed || i &gt; abs(length)          % line search failed twice in a row
break;                             % or we ran out of time, so we give up
end
tmp = df1; df1 = df2; df2 = tmp;                         % swap derivatives
s = -df1;                                                    % try steepest
d1 = -s'*s;
z1 = 1/(1-d1);
ls_failed = 1;                                    % this line search failed
end
if exist('OCTAVE_VERSION')
fflush(stdout);
end
end
fprintf('\n');
```

predict.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);

h1 = sigmoid([ones(m, 1) X] * Theta1');
h2 = sigmoid([ones(m, 1) h1] * Theta2');
[dummy, p] = max(h2, [], 2);

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

end
```

sigmoid.m

```function g = sigmoid(z)
%SIGMOID Compute sigmoid functoon
%   J = SIGMOID(z) computes the sigmoid of z.

g = 1.0 ./ (1.0 + exp(-z));
end
```

submit.m

```function submit()

conf.assignmentSlug = 'neural-network-learning';
conf.itemName = 'Neural Networks Learning';
conf.partArrays = { ...
{ ...
'1', ...
{ 'nnCostFunction.m' }, ...
'Feedforward and Cost Function', ...
}, ...
{ ...
'2', ...
{ 'nnCostFunction.m' }, ...
'Regularized Cost Function', ...
}, ...
{ ...
'3', ...
}, ...
{ ...
'4', ...
{ 'nnCostFunction.m' }, ...
}, ...
{ ...
'5', ...
{ 'nnCostFunction.m' }, ...
}, ...
};
conf.output = @output;

submitWithConfiguration(conf);
end

function out = output(partId, auxstring)
% Random Test Cases
X = reshape(3 * sin(1:1:30), 3, 10);
Xm = reshape(sin(1:32), 16, 2) / 5;
ym = 1 + mod(1:16,4)';
t1 = sin(reshape(1:2:24, 4, 3));
t2 = cos(reshape(1:2:40, 4, 5));
t  = [t1(:) ; t2(:)];
if partId == '1'
[J] = nnCostFunction(t, 2, 4, 4, Xm, ym, 0);
out = sprintf('%0.5f ', J);
elseif partId == '2'
[J] = nnCostFunction(t, 2, 4, 4, Xm, ym, 1.5);
out = sprintf('%0.5f ', J);
elseif partId == '3'
elseif partId == '4'
[J, grad] = nnCostFunction(t, 2, 4, 4, Xm, ym, 0);
out = sprintf('%0.5f ', J);
out = [out sprintf('%0.5f ', grad)];
elseif partId == '5'
[J, grad] = nnCostFunction(t, 2, 4, 4, Xm, ym, 1.5);
out = sprintf('%0.5f ', J);
out = [out sprintf('%0.5f ', grad)];
end
end

```

ex4data1.mat

```jar:file:/Applications/MATLAB_R2015b.app/java/jar/common.jar!/com/mathworks/common/icons/resources/variable_matrix.png	X	5000x400 double
jar:file:/Applications/MATLAB_R2015b.app/java/jar/common.jar!/com/mathworks/common/icons/resources/variable_matrix.png	y	5000x1 double
```

ex4weights.mat

```jar:file:/Applications/MATLAB_R2015b.app/java/jar/common.jar!/com/mathworks/common/icons/resources/variable_matrix.png	Theta1	25x401 double
jar:file:/Applications/MATLAB_R2015b.app/java/jar/common.jar!/com/mathworks/common/icons/resources/variable_matrix.png	Theta2	10x26 double
```

[ ex4.m Output ]

```Loading and Visualizing Data ...
Program paused. Press enter to continue.
```

Figure 1 (Examples from the dataset)

```Loading Saved Neural Network Parameters ...

Feedforward Using Neural Network ...
Cost at parameters (loaded from ex4weights): 0.287629
(this value should be about 0.287629)

Program paused. Press enter to continue.
```
```Checking Cost Function (w/ Regularization) ...
Cost at parameters (loaded from ex4weights): 0.383770
(this value should be about 0.383770)
Program paused. Press enter to continue.
```
```
Sigmoid gradient evaluated at [1 -0.5 0 0.5 1]:
0.196612 0.235004 0.250000 0.235004 0.196612

Program paused. Press enter to continue.
```
```Initializing Neural Network Parameters ...

Checking Backpropagation...
-0.0093   -0.0093
0.0089    0.0089
-0.0084   -0.0084
0.0076    0.0076
-0.0067   -0.0067
-0.0000   -0.0000
0.0000    0.0000
-0.0000   -0.0000
0.0000    0.0000
-0.0000   -0.0000
-0.0002   -0.0002
0.0002    0.0002
-0.0003   -0.0003
0.0003    0.0003
-0.0004   -0.0004
-0.0001   -0.0001
0.0001    0.0001
-0.0001   -0.0001
0.0002    0.0002
-0.0002   -0.0002
0.3145    0.3145
0.1111    0.1111
0.0974    0.0974
0.1641    0.1641
0.0576    0.0576
0.0505    0.0505
0.1646    0.1646
0.0578    0.0578
0.0508    0.0508
0.1583    0.1583
0.0559    0.0559
0.0492    0.0492
0.1511    0.1511
0.0537    0.0537
0.0471    0.0471
0.1496    0.1496
0.0532    0.0532
0.0466    0.0466

The above two columns you get should be very similar.

If your backpropagation implementation is correct, then
the relative difference will be small (less than 1e-9).

Relative Difference: 2.35344e-11

Program paused. Press enter to continue.
```
```Cost at (fixed) debugging parameters (w/ lambda = 10): 0.576051
(this value should be about 0.576051)

Program paused. Press enter to continue.
```
```Training Neural Network...
Iteration     1 | Cost: 3.309131e+00
Iteration     2 | Cost: 3.260624e+00
Iteration     3 | Cost: 3.213711e+00
Iteration     4 | Cost: 3.088849e+00
Iteration     5 | Cost: 2.827554e+00
Iteration     6 | Cost: 2.212024e+00
Iteration     7 | Cost: 1.988141e+00
Iteration     8 | Cost: 1.926129e+00
Iteration     9 | Cost: 1.802847e+00
Iteration    10 | Cost: 1.647500e+00
Iteration    11 | Cost: 1.589213e+00
Iteration    12 | Cost: 1.452209e+00
Iteration    13 | Cost: 1.341458e+00
Iteration    14 | Cost: 1.242945e+00
Iteration    15 | Cost: 1.134786e+00
Iteration    16 | Cost: 1.016537e+00
Iteration    17 | Cost: 9.485571e-01
Iteration    18 | Cost: 9.138379e-01
Iteration    19 | Cost: 8.725694e-01
Iteration    20 | Cost: 8.343245e-01
Iteration    21 | Cost: 7.921763e-01
Iteration    22 | Cost: 7.690259e-01
Iteration    23 | Cost: 7.610273e-01
Iteration    24 | Cost: 7.354194e-01
Iteration    25 | Cost: 7.218934e-01
Iteration    26 | Cost: 7.159318e-01
Iteration    27 | Cost: 7.012427e-01
Iteration    28 | Cost: 6.920974e-01
Iteration    29 | Cost: 6.812728e-01
Iteration    30 | Cost: 6.505659e-01
Iteration    31 | Cost: 6.205947e-01
Iteration    32 | Cost: 6.003176e-01
Iteration    33 | Cost: 5.887892e-01
Iteration    34 | Cost: 5.747566e-01
Iteration    35 | Cost: 5.687621e-01
Iteration    36 | Cost: 5.627345e-01
Iteration    37 | Cost: 5.541156e-01
Iteration    38 | Cost: 5.508178e-01
Iteration    39 | Cost: 5.485831e-01
Iteration    40 | Cost: 5.438776e-01
Iteration    41 | Cost: 5.415809e-01
Iteration    42 | Cost: 5.333668e-01
Iteration    43 | Cost: 5.189307e-01
Iteration    44 | Cost: 5.117082e-01
Iteration    45 | Cost: 5.070935e-01
Iteration    46 | Cost: 4.973425e-01
Iteration    47 | Cost: 4.923056e-01
Iteration    48 | Cost: 4.858475e-01
Iteration    49 | Cost: 4.793730e-01
Iteration    50 | Cost: 4.750988e-01

Program paused. Press enter to continue.
```
```Visualizing Neural Network...

Program paused. Press enter to continue.
```

Figure 2 (Visualization of Hidden Units.)

```//result:

Training Set Accuracy: 95.560000
```

[ Unit Testing ]

```sigmoidGradient([[-1 -2 -3] ; magic(3)])
ans =
1.9661e-001  1.0499e-001  4.5177e-002
3.3524e-004  1.9661e-001  2.4665e-003
4.5177e-002  6.6481e-003  9.1022e-004
1.7663e-002  1.2338e-004  1.0499e-001

```
```u = sigmoid([[-1 -2 -3] ; magic(3)])
% result
u =
0.268941   0.119203   0.047426
0.999665   0.731059   0.997527
0.952574   0.993307   0.999089
0.982014   0.999877   0.880797

```
```u.*(1-u)
% result
ans =
1.9661e-001  1.0499e-001  4.5177e-002
3.3524e-004  1.9661e-001  2.4665e-003
4.5177e-002  6.6481e-003  9.1022e-004
1.7663e-002  1.2338e-004  1.0499e-001

```

nnCostFunction() with (and without) regularization:
Enter these values in your console workspace,
compare your results with those given.

Test Case with regularization:

```il = 2;              % input layer
hl = 2;              % hidden layer
nl = 4;              % number of labels
nn = [ 1:18 ] / 10;  % nn_params
X = cos([1 2 ; 3 4 ; 5 6]);
y = [4; 2; 3];
lambda = 4;
[J grad] = nnCostFunction(nn, il, hl, nl, X, y, lambda)
```

output:

```J = 19.474
0.76614
0.97990
0.37246
0.49749
0.64174
0.74614
0.88342
0.56876
0.58467
0.59814
1.92598
1.94462
1.98965
2.17855
2.47834
2.50225
2.52644
2.72233
```

Here are the values for all internal variables for the regularized test case:

```d2 =
0.79393   1.05281
0.73674   0.95128
0.76775   0.93560

d3 =
0.888659   0.907427   0.923305  -0.063351
0.838178  -0.139718   0.879800   0.896918
0.923414   0.938578  -0.049102   0.960851

Delta1 =
2.298415  -0.082619  -0.074786
2.939691  -0.107533  -0.161585

Delta2 =
2.65025   1.37794   1.43501
1.70629   1.03385   1.10676
1.75400   0.76894   0.77931
1.79442   0.93566   0.96699

z2 =
0.054017   0.166433
-0.523820  -0.588183
0.665184   0.889567

ans =
0.24982   0.24828
0.23361   0.22957
0.22426   0.20640

a2 =
1.00000   0.51350   0.54151
1.00000   0.37196   0.35705
1.00000   0.66042   0.70880

a3 =
0.88866   0.90743   0.92330   0.93665
0.83818   0.86028   0.87980   0.89692
0.92341   0.93858   0.95090   0.96085

```

——————————–

Test case without regularization (uses same data, but 0 for lambda):

```>> [J grad] = nnCostFunction(nn, il, hl, nl, X, y, 0)
J =  7.4070
0.766138
0.979897
-0.027540
-0.035844
-0.024929
-0.053862
0.883417
0.568762
0.584668
0.598139
0.459314
0.344618
0.256313
0.311885
0.478337
0.368920
0.259771
0.322331

```

============

Values for Delta1 and Delta2 (the unregularized gradient, from tutorial Step 5 and Step 6) – truncated to 3 decimal places, prior to scaling by 1/m.

```Delta1 =
2.298 -0.082 -0.074
2.939 -0.107 -0.161

Delta2 =
2.650  1.377  1.435
1.706  1.033  1.106
1.754  0.768  0.779
1.794  0.935  0.966

```

## Author: iotmaker

I am interested in IoT, robot, figures & leadership. Also, I have spent almost every day of the past 15 years making robots or electronic inventions or computer programs.