CheeseZH: Stanford University: Machine Learning Ex4:Training Neural Network(Backpropagation Algorithm)

1. Feedforward and cost function;

2.Regularized cost function:

3.Sigmoid gradient

The gradient for the sigmoid function can be computed as:

where:

4.Random initialization

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 "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 row of W corresponds to the parameters for the bias units
%
epsilon_init = 0.12;
W = rand(L_out, 1 + L_in) * 2 * epsilon_init - epsilon_init;

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

end

 

5.Backpropagation(using a for-loop for t=1:m and place steps 1-4 below inside the for-loop), with the t^th iteration perfoming the calculation on the t^th training example(x^(t),y^(t)).Step 5 will divide the accumulated gradients by m to obtain the gradients for the neural network cost function.

(1) Set the input layer's values(a⁽¹⁾) to the t-th training example x^(t). Perform a feedforward pass, computing the activations(z⁽²⁾,a⁽²⁾,z⁽³⁾,a⁽³⁾) for layers 2 and 3.

(2) For each output unit k in layer 3(the output layer), set :

where y_k = 1 or 0.

(3)For the hidden layer l=2, set:

(4) Accumulate the gradient from this example using the following formula. Note that you should skip or remove δ₀(2).

(5) Obtain the(unregularized) gradient for the neural network cost function by dividing the accumulated gradients by 1/m:

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);
         
% 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.
%

%Part 1
%Theta1 has size 25*401
%Theta2 has size 10*26
%y hase size 5000*1
K = num_labels;
Y = eye(K)(y,:); %[5000 10]
a1 = [ones(m,1),X];%[5000 401]
a2 = sigmoid(a1*Theta1'); %[5000 25]
a2 = [ones(m,1),a2];%[5000 26]
h = sigmoid(a2*Theta2');%[5000 10]

costPositive = -Y.*log(h);
costNegtive = (1-Y).*log(1-h); 
cost = costPositive - costNegtive;
J = (1/m)*sum(cost(:));
%Regularized
Theta1Filtered = Theta1(:,2:end); %[25 400]
Theta2Filtered = Theta2(:,2:end); %[10 25]
reg = (lambda/(2*m))*(sumsq(Theta1Filtered(:))+sumsq(Theta2Filtered(:)));
J = J + reg;


%Part 2
Delta1 = 0;
Delta2 = 0;
for t=1:m,
  %step 1
  a1 = [1 X(t,:)]; %[1 401]
  z2 = a1*Theta1'; %[1 25]
  a2 = [1 sigmoid(z2)];%[1 26]
  z3 = a2*Theta2'; %[1 10]
  a3 = sigmoid(z3); %[1 10]
  %step 2
  yt = Y(t,:);%[1 10]
  d3 = a3-yt; %[1 10]
  %step 3
  %   [1 10]  [10 25]           [1 25]
  d2 = (d3*Theta2Filtered).*sigmoidGradient(z2); %[1 25]
  %step 4
  Delta1 = Delta1 + (d2'*a1);%[25 401]
  Delta2 = Delta2 + (d3'*a2);%[10 26]  
end;

%step 5
Theta1_grad = (1/m)*Delta1;
Theta2_grad = (1/m)*Delta2;

%Part 3
Theta1_grad(:,2:end) = Theta1_grad(:,2:end) + ((lambda/m)*Theta1Filtered);
Theta2_grad(:,2:end) = Theta2_grad(:,2:end) + ((lambda/m)*Theta2Filtered);

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

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

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


end

 

6.Gradient checking

Let

and

for each i, that:

computeNumericalGradient.m

function numgrad = computeNumericalGradient(J, theta)
%COMPUTENUMERICALGRADIENT Computes the gradient using "finite differences"
%and gives us a numerical estimate of the gradient.
%   numgrad = COMPUTENUMERICALGRADIENT(J, theta) computes the numerical
%   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 
%        returns the numerical gradient.It sets numgrad(i) to (a numerical 
%        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).)
%                

numgrad = zeros(size(theta));
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);
    % Compute Numerical Gradient
    numgrad(p) = (loss2 - loss1) / (2*e);
    perturb(p) = 0;
end

end

 

7.Regularized Neural Networks

for j=0:

for j>=1:

 

别人的代码：

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