Stanford coursera Andrew Ng 机器学习课程第四周总结（附Exercise 3）

Introduction

Neural NetWork的由来

先考虑一个非线性分类，当特征数很少时，逻辑回归就可以完成了，但是当特征数变大时，高阶项将呈指数性增长，复杂度可想而知。如下图：对房屋进行高低档的分类，当特征值只有x1,x2,x3时，我们可以对它进行处理，分类。但是当特征数增长为x1,x2....x100时，分类器的效率就会很低了。 

Neural NetWork模型

 

该图是最简单的神经网络，共有3层，输入层Layer1；隐藏层Layer2；输出层Layer3，每层都有多个激励函数ai(j).通过层与层之间的传递参数Θ得到最终的假设函数hΘ(x)。我们的目的是通过大量的输入样本x（作为第一层），训练层与层之间的传递参数(经常称为权重)，使得假设函数尽可能的与实际输出值接近h(x)≈y(代价函数J尽可能的小)。

逻辑回归模型

很容易看出，逻辑回归是没有隐藏层的神经网络，层与层之间的传递函数就是θ。

Neural NetWork

神经网络模型---正向传播  

 

Cost function(代价函数)

Examples and intuitions

 

Multi-class classification

对于多分类问题，我们可以通过设置多个输出值来实现。

编程作业就是一个多分类问题——手写数字识别

输入的是手写的照片（数字0-9），5000组样本、每个像素点用20×20的点阵表示成一行，输入向量为5000×400的矩阵X，经过神经网络传递后，输出一个假设函数(列向量),取最大值所在的行号即为假设值(0-9中的一个)。也就是输出值y = 1，2，3，4，5.....10又有可能，为了方便数值运算，我们用10×1的列向量表示，譬如 y = 5，有

Exercises

这次的作业是用逻辑回归和神经网络来实现手写数字识别，比较下两者的准确性。

Logistic Regression

lrCostFunction.m

function [J, grad] = lrCostFunction(theta, X, y, lambda)
%LRCOSTFUNCTION Compute cost and gradient for logistic regression with 
%regularization
%   J = LRCOSTFUNCTION(theta, X, y, lambda) computes the cost of using
%   theta as the parameter for regularized logistic regression and the
%   gradient of the cost w.r.t. to the parameters. 

% Initialize some useful values
m = length(y); % number of training examples

% You need to return the following variables correctly 
J = 0;
grad = zeros(size(theta));

% ====================== YOUR CODE HERE ======================
% Instructions: Compute the cost of a particular choice of theta.
%               You should set J to the cost.
%               Compute the partial derivatives and set grad to the partial
%               derivatives of the cost w.r.t. each parameter in theta
%
% Hint: The computation of the cost function and gradients can be
%       efficiently vectorized. For example, consider the computation
%
%           sigmoid(X * theta)
%
%       Each row of the resulting matrix will contain the value of the
%       prediction for that example. You can make use of this to vectorize
%       the cost function and gradient computations. 
%
% Hint: When computing the gradient of the regularized cost function, 
%       there're many possible vectorized solutions, but one solution
%       looks like:
%           grad = (unregularized gradient for logistic regression)
%           temp = theta; 
%           temp(1) = 0;   % because we don't add anything for j = 0  
%           grad = grad + YOUR_CODE_HERE (using the temp variable)
%
theta_reg=[0;theta(2:size(theta))];

J = (-y'*log(sigmoid(X*theta))-(1-y)'*log(1-sigmoid(X*theta)))/m + lambda/(2*m)*(theta_reg')*theta_reg;

grad = X'*(sigmoid(X*theta)-y)/m + lambda/m*theta_reg;

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

grad = grad(:);

end

oneVsAll.m 

function [all_theta] = oneVsAll(X, y, num_labels, lambda)
%ONEVSALL trains multiple logistic regression classifiers and returns all
%the classifiers in a matrix all_theta, where the i-th row of all_theta 
%corresponds to the classifier for label i
%   [all_theta] = ONEVSALL(X, y, num_labels, lambda) trains num_labels
%   logistic regression classifiers and returns each of these classifiers
%   in a matrix all_theta, where the i-th row of all_theta corresponds 
%   to the classifier for label i

% Some useful variables
m = size(X, 1);
n = size(X, 2);

% You need to return the following variables correctly 
all_theta = zeros(num_labels, n + 1);

% Add ones to the X data matrix
X = [ones(m, 1) X];

% ====================== YOUR CODE HERE ======================
% Instructions: You should complete the following code to train num_labels
%               logistic regression classifiers with regularization
%               parameter lambda. 
%
% Hint: theta(:) will return a column vector.
%
% Hint: You can use y == c to obtain a vector of 1's and 0's that tell you
%       whether the ground truth is true/false for this class.
%
% Note: For this assignment, we recommend using fmincg to optimize the cost
%       function. It is okay to use a for-loop (for c = 1:num_labels) to
%       loop over the different classes.
%
%       fmincg works similarly to fminunc, but is more efficient when we
%       are dealing with large number of parameters.
%
% Example Code for fmincg:
%
%     % Set Initial theta
%     initial_theta = zeros(n + 1, 1);
%     
%     % Set options for fminunc
%     options = optimset('GradObj', 'on', 'MaxIter', 50);
% 
%     % Run fmincg to obtain the optimal theta
%     % This function will return theta and the cost 
%     [theta] = ...
%         fmincg (@(t)(lrCostFunction(t, X, (y == c), lambda)), ...
%                 initial_theta, options);
%


 initial_theta = zeros(n + 1, 1);
 
 options = optimset('GradObj', 'on', 'MaxIter', 50);
 
 for c = 1:num_labels
    all_theta(c,:) = fmincg (@(t)(lrCostFunction(t, X, (y == c), lambda)), initial_theta, options);
 end



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


end

predictOneVsAll.m

function p = predictOneVsAll(all_theta, X)
%PREDICT Predict the label for a trained one-vs-all classifier. The labels 
%are in the range 1..K, where K = size(all_theta, 1). 
%  p = PREDICTONEVSALL(all_theta, X) will return a vector of predictions
%  for each example in the matrix X. Note that X contains the examples in
%  rows. all_theta is a matrix where the i-th row is a trained logistic
%  regression theta vector for the i-th class. You should set p to a vector
%  of values from 1..K (e.g., p = [1; 3; 1; 2] predicts classes 1, 3, 1, 2
%  for 4 examples) 

m = size(X, 1);
num_labels = size(all_theta, 1);

% You need to return the following variables correctly 
p = zeros(size(X, 1), 1);

% Add ones to the X data matrix
X = [ones(m, 1) X];

% ====================== YOUR CODE HERE ======================
% Instructions: Complete the following code to make predictions using
%               your learned logistic regression parameters (one-vs-all).
%               You should set p to a vector of predictions (from 1 to
%               num_labels).
%
% Hint: This code can be done all vectorized using the max function.
%       In particular, the max function can also return the index of the 
%       max element, for more information see 'help max'. If your examples 
%       are in rows, then, you can use max(A, [], 2) to obtain the max 
%       for each row.
%       

[maxx, p]=max(X*all_theta',[],2);


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


end

Training Set Accuracy: 95.100000

下面是以三层bp神经网络处理的手写数字识别，其中权重矩阵已给出。

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);

% ====================== YOUR CODE HERE ======================
% Instructions: Complete the following code to make predictions using
%               your learned neural network. You should set p to a 
%               vector containing labels between 1 to num_labels.
%
% Hint: The max function might come in useful. In particular, the max
%       function can also return the index of the max element, for more
%       information see 'help max'. If your examples are in rows, then, you
%       can use max(A, [], 2) to obtain the max for each row.
%
X = [ones(m, 1) X];

temp=sigmoid(X*Theta1');

temp = [ones(m, 1) temp];

temp2=sigmoid(temp*Theta2');

[maxx, p]=max(temp2, [], 2);


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


end

Training Set Accuracy: 97.520000

注意事项

1.X = [ones(m, 1) X];是确保矩阵维度一致。X0就是一行1

2.正则化时theta0要用0替代，处理如theta_reg=[0;theta(2:size(theta))];