【题解】编程作业ex3: Multi-class Classification and Neural Networks (Machine Learning)

吐槽：有点点难，但可以推出的。。因为感觉都值得写所以就都写了，顺便说了说思路，如果有更好的思路也可以评论我hhh

题目：

Download the programming assignment here.

This ZIP file contains the instructions in a PDF and the starter code. You may use either MATLAB or Octave (>= 3.8.0). To submit this assignment, call the included submit function from MATLAB / Octave. You will need to enter the token provided on the right-hand side of this page.

lrCostFunction我的解法：

pdf在这里提示了两个点，一个是向量法的输出可以用size维度来检测其正确性，另一个是可以用theta(2:end)切片且用.^2来做element-wise的操作。我觉得需要注意的还是theta0是不需要lambda改变的，所以无论J还是grad都需要从theta1开始考虑，这个在代码里面也有hint。

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)
%
h = sigmoid(X * theta);
J = 1/m * (-y'*log(h) - (1-y)'*log(1-h)) + lambda/(2*m) * sum(theta(2:end).^2);
grad = 1/m * X' * (sigmoid(X * theta) - y);
temp = theta;
temp(1) = 0;
grad = grad + lambda/m * temp;

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

grad = grad(:);

end

oneVsAll我的解法：

这个函数本来我有点没理解，但是翻看了笔记里面对one-vs-all的定义，h^(i)(x)是对于第 i 个class概率，然后max(h^(i)(x))处 i 的取值即为分类结果，所以每个h(x)都有一组theta，i个h(x)有 i 组theta。而且代码中的注释里：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，意思就是第 i 组theta需要放在第 i 行all_theta里面，因此需要转置一下。而在pdf里面的tips的代码运行后发现返回的是个和 a 维度一样的只有0和1组成的代表真假的矩阵，所以y==c中的c也只是常数，不是一个向量。

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);
%
for c = 1:num_labels,
  % 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);
  
  % Set theta to the c-th row in all_theta
  all_theta(c, :) = theta';
  
endfor

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

end
predictOneVsAll我的解法：

一开始觉得看这个描述似乎很复杂的样子，而且题目还提示说from 1 to num_labels，于是尝试了一下用for循环做这个，但是没有成功，感觉太过于繁琐了。然后又查了一下max(A, [], 2)这个语法的含义是取每一行的最大值（https://www.cnblogs.com/liuxjie/p/12024942.html），于是思路改变一下可能就是要求出某个矩阵然后求每一行的最大值，那么看一下维度，all_theta是 i * (n+1)，X是 m * (n+1)，而返回值 p 是 m*1 ，所以自然的可以知道中间矩阵A是 g(X*all_theta')。

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.
%       
A = sigmoid(X * all_theta');
[x, p] = max(A, [], 2);

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

end

predict我的解法：

分析一下维度发现就是这么做的=。=不过需要注意一下octave里面似乎不支持多维矩阵哎，所以得写成A1A2A3这种形式。。

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.
%

% Add ones to the X data matrix
X = [ones(m, 1) X];
A1 = X;
A2 = [ones(m, 1) sigmoid(A1 * Theta1')];
A3 = sigmoid(A2 * Theta2');
[x, p] = max(A3, [], 2);

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

end