Coursera机器学习课程笔记（五）

Coursera机器学习课程笔记（五）

由于好几题都是在nnCostFunction.m这一个文件中完成，所以一并给出1、2、4和5的代码

1.Feedforward and Cost Function

2.Regularized Cost Function

4.Neural Network Gradient (Backpropagation)

5.Regularized Gradient

nnCostFunction.m文件：

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.
%

%模拟神经网络的前向传播过程，Theta1和Theta2是已经训练好的
a1 = X;           %第一层

a1 = [ones(m,1),a1];
z2 = a1 * Theta1';
a2 = sigmoid(z2);       %第二层

a2 = [ones(m,1),a2];
z3 = a2 * Theta2';
a3 = sigmoid(z3);       %第三层，a3可看作是输出结果h(x)

for k = 1:num_labels
    %根据公式，计算当k为不同数字时的误差之和，通过a3(:,k)来取出对应的列与(y==k)矩阵进行运算
    J = J + (1/m) * (-(y == k)' * log(a3(:,k)) - (1-(y == k))' * log(1 - a3(:,k)));
end

%下面单独计算正则化的尾项
sum = 0;
for i = 1:hidden_layer_size
    for j = 2:input_layer_size + 1        % Theta1第一列不计算
        sum = sum + Theta1(i,j)^2;
    end
end

for i = 1:num_labels
    for j = 2:hidden_layer_size + 1      %Theta2第一列不计算
        sum = sum + Theta2(i,j)^2;
    end
end

J = J + lambda/(2*m) * sum;       %包含正则化的损失函数J

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

%反向传播算法
delta_1 = 0;
delta_2 = 0;
for t = 1:m      %根据要求，分成m次循环实现反向传播，m代表训练样本的个数 
    d_3 = a3(t,:)' - ([1:num_labels] == y(t,1))';         %对应step2，d_3为10x1矩阵
    d_2 = Theta2(:,2:end)' * d_3 .* sigmoidGradient(z2(t,:)'); %对应step3，注意Theta2的第一列为bias所以舍去，d_2为25x1矩阵
    delta_1 = delta_1 + d_2 * a1(t,:);   %对应step4
    delta_2 = delta_2 + d_3 * a2(t,:);   %对应step4
end

%包含正则化的梯度
Theta1(:,1) = 0;        %把第一列全置0，不需要考虑正则化
Theta1_grad = delta_1 / m + lambda / m * Theta1;
Theta2(:,1) = 0;
Theta2_grad = delta_2 / m + lambda / m * Theta2;

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

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


end

关于反向传播算法这一部分的代码，强烈建议在for循环处建立断点，在调试过程观察各个矩阵的维度，就能理解代码在做什么了。

3.Sigmoid Gradient

sigmoidGradient.m文件：

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 = exp(-z) ./ (1 + exp(-z)).^2;

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

end