week4 编程作业回顾

这一周课程主要内容是关于神经网络初步，包括一些逻辑运算结构等等，总体难度不大，但是编程还是有很多需要注意的地方。

下面附上我的代码以及我在编程过程中出现的问题：

这个作业总的要求是让你根据5000 个 20 ＊20 的灰度矩阵来识别字体

1. 首先是logical regression with regularization 这跟这周讲的其实没有多大联系：

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 = -1/m*(y'*log(sigmoid(X*theta))+(1-y')*log(1-sigmoid(X*theta)))+lambda/(2*m)*(sum(theta.^2)-theta(1)*theta(1));
%grad = 1/m*X'*(X*theta-y')+lambda/m*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)
%

grad = 1/m*X'*(sigmoid(X*theta)-y);
temp = theta;
temp(1)=0;
grad = grad + (lambda/m)*temp;

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


end

重点是logical regression的表达式有那个表达式这个函数就可以轻松的写出来，附的pdf 上面写的非常清楚：

以上就是代价函数的表达式用python的语言来表达也十分容易

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

因为我们用的是一些标准化的方法所以在cost这个函数中我们也必须要给出目前的梯度供 fmincg 这个函数使用

有了代价函数我们就很容易可以推导出偏导数：

也是非常容易表示

grad = 1/m*X'*(sigmoid(X*theta)-y);
temp = theta;
temp(1)=0;
grad = grad + (lambda/m)*temp;

以上两个地方都需要特别注意的一点是在进行运算中 theta(1) 并不会参与到运算中所以theta(1) = 0

2. onevsall

这个函数的实质是对于我们的parameters 进行训练

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

在这里我们需要注意的是fmincg 这个函数的使用方法：

在其中第一个参数是我们需要最小化的函数在这里是lrCostFunction

[jVal,gradient] 注意这个函数返回两个值，虽然这里没用但是也值得一说第一个经过迭代以后的的代价当这个代价比较小的时候我们这次训练才是有意义的。第二个就是梯度，相当于我们训练的能够优化代价函数的一些参数，在这里也是我们训练的一些参数。

训练之后注意赋给kk 由kk再赋给all_theta (我的matlab语言学的不怎么样如果有更好的方法请一定联系我)

在这里训练的时候只能够是向量训练，所以我们把 10 *401 的矩阵化成1*401 的向量进行分别训练最后再合在一起。

3.onevsall predict

这个就非常简单了并且max函数根据介绍也能返回最大值的索引所以针对每一个行向量用一次max 或者根据max函数的第二个返回值去做就好，详情请baidu或google octave中max函数的用法

4.predict

这个就更简单了训练好了神经网络要求预测值和 3基本上一样，下面上核心代码

for i = 1:m
  p(i)=find(temp(i,:)==max(temp(i,:)))

week4 编程作业 回顾

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