Stanford coursera Andrew Ng 機器學習課程第四周總結（附Exercise 3）

Introduction

Neural NetWork的由來

先考慮一個非線性分類，當特徵數不多時，邏輯迴歸就能夠完成了，可是當特徵數變大時，高階項將呈指數性增加，複雜度可想而知。以下圖：對房屋進行高低檔的分類，當特徵值只有x1,x2,x3時，咱們能夠對它進行處理，分類。可是當特徵數增加爲x1,x2....x100時，分類器的效率就會很低了。 

Neural NetWork模型

 

 

該圖是最簡單的神經網絡，共有3層，輸入層Layer1；隱藏層Layer2；輸出層Layer3，每層都有多個激勵函數ai(j).經過層與層之間的傳遞參數Θ獲得最終的假設函數hΘ(x)。咱們的目的是經過大量的輸入樣本x（做爲第一層），訓練層與層之間的傳遞參數(常常稱爲權重)，使得假設函數儘量的與實際輸出值接近h(x)≈y(代價函數J儘量的小)。

邏輯迴歸模型

 

很容易看出，邏輯迴歸是沒有隱藏層的神經網絡，層與層之間的傳遞函數就是θ。

Neural NetWork

神經網絡模型---正向傳播  

 

 

Cost function(代價函數)

Examples and intuitions

 

 

 

 

 

 

 

 

 

 

 

 

 

Multi-class classification

對於多分類問題，咱們能夠經過設置多個輸出值來實現。

編程做業就是一個多分類問題——手寫數字識別

輸入的是手寫的照片（數字0-9），5000組樣本、每一個像素點用20×20的點陣表示成一行，輸入向量爲5000×400的矩陣X，通過神經網絡傳遞後，輸出一個假設函數(列向量),取最大值所在的行號即爲假設值(0-9中的一個)。也就是輸出值y = 1，2，3，4，5.....10又有可能，爲了方便數值運算，咱們用10×1的列向量表示，譬如 y = 5，有

 

Exercises

此次的做業是用邏輯迴歸和神經網絡來實現手寫數字識別，比較下二者的準確性。

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

 

下面是以三層bp神經網絡處理的手寫數字識別，其中權重矩陣已給出。

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

注意事項

1.X = [ones(m, 1) X];是確保矩陣維度一致。X0就是一行1

2.正則化時theta0要用0替代，處理如theta_reg=[0;theta(2:size(theta))];