Stanford机器学习笔记-5.神经网络Neural Networks (part two)

5 Neural Networks (part two)

content:

　　5 Neural Networks (part two)

　　　　5.1 cost function

　　　　5.2 Back Propagation

　　　　5.3 神经网络总结

接上一篇4. Neural Networks (part one). 本文将先定义神经网络的代价函数，然后介绍逆向传播(Back Propagation: BP)算法，它能有效求解代价函数对连接权重的偏导，最后对训练神经网络的过程进行总结。

5.1 cost function

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: 1->2; Theta2: 2->3
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);
J = 0;
Theta1_grad = zeros(size(Theta1));
Theta2_grad = zeros(size(Theta2));

%         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.

for i = 1:m
% compute activation by Forward Propagation
a1 = [1; X(i,:)'];
z2 = Theta1 * a1;
a2 = [1; sigmoid(z2)];
z3 = Theta2 * a2;
h = sigmoid(z3);

yy = zeros(num_labels,1);
yy(y(i)) = 1;              % 训练集的真实值yy

J = J + sum(-yy .* log(h) - (1-yy) .* log(1-h));

% Back Propagation
delta3 = h - yy;
delta2 = (Theta2(:,2:end)' * delta3) .* sigmoidGradient(z2); %注意要除去偏移单元的连接权重

Theta2_grad = Theta2_grad + delta3 * a2';
Theta1_grad = Theta1_grad + delta2 * a1';
end

J = J / m + lambda * (sum(sum(Theta1(:,2:end) .^ 2)) + sum(sum(Theta2(:,2:end) .^ 2))) / (2*m);

Theta2_grad = Theta2_grad / m;
Theta2_grad(:,2:end) = Theta2_grad(:,2:end) + lambda * Theta2(:,2:end) / m; % regularized nn

Theta1_grad = Theta1_grad / m;
Theta1_grad(:,2:end) = Theta1_grad(:,2:end) + lambda * Theta1(:,2:end) / m; % regularized nn

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

end

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