吐槽：有點點難，但可以推出的。。因為感覺都值得寫是以就都寫了，順便說了說思路，如果有更好的思路也可以評論我hhh
題目：
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在這裡提示了兩個點，一個是向量法的輸出可以用size次元來檢測其正确性，另一個是可以用theta(2:end)切片且用.^2來做element-wise的操作。我覺得需要注意的還是theta0是不需要lambda改變的，是以無論J還是grad都需要從theta1開始考慮，這個在代碼裡面也有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：
這個函數本來我有點沒了解，但是翻看了筆記裡面對one-vs-all的定義，h^(i)(x)是對于第 i 個class機率，然後max(h^(i)(x))處 i 的取值即為分類結果，是以每個h(x)都有一組theta，i個h(x)有 i 組theta。而且代碼中的注釋裡：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 組theta需要放在第 i 行all_theta裡面，是以需要轉置一下。而在pdf裡面的tips的代碼運作後發現傳回的是個和 a 次元一樣的隻有0和1組成的代表真假的矩陣，是以y==c中的c也隻是常數，不是一個向量。
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：
一開始覺得看這個描述似乎很複雜的樣子，而且題目還提示說from 1 to num_labels，于是嘗試了一下用for循環做這個，但是沒有成功，感覺太過于繁瑣了。然後又查了一下max(A, [], 2)這個文法的含義是取每一行的最大值（https://www.cnblogs.com/liuxjie/p/12024942.html），于是思路改變一下可能就是要求出某個矩陣然後求每一行的最大值，那麼看一下次元，all_theta是 i * (n+1)，X是 m * (n+1)，而傳回值 p 是 m*1 ，是以自然的可以知道中間矩陣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：
分析一下次元發現就是這麼做的=。=不過需要注意一下octave裡面似乎不支援多元矩陣哎，是以得寫成A1A2A3這種形式。。
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
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