前言
學以緻用，以學促用，通過筆記總結，鞏固學習成果，複習新學的概念。
目錄
文章目錄
- 前言
- 目錄
- 正文
-
- 模型引入
- 神經網絡
- 模型表示
-
- 模型表示2
- 例子和圖示
- 例子與圖示2
- 作業答案
正文
本節主要讨論神經網絡及其強大的功能
模型引入
非線性分類問題會帶來超多的參數，也就是參數爆炸這一問題。
一個典型的例子，計算機視覺目辨別别問題。
通過訓練分類器，我們可以很好的确定一個像素對應的是不是汽車。
神經網絡
神經網路的發展曆史。
神經網絡的誘因，大腦并沒有特定的神經元去處理特定的任務。
模型表示
神經元模型結構圖
神經元模型數學化，邏輯單元。
神經元模型，多層感覺機。
多層神經網絡的數學表示和參數量。
模型表示2
神經網絡向量化實作方法，前向傳遞計算流程。
神經網絡學到了它的特征。
其他類似的神經網絡架構。
例子和圖示
非線性分類問題：異或。
簡化例子，邏輯與操作
簡化例子，或操作。
例子與圖示2
簡單例子，否操作的實作。
簡單例子：異或操作。
應用例子：手寫文本識别。
## 多分類問題
多目辨別别問題，通過onehot編碼，輸出圖像裡的各個類别。
輸出層也有多個神經元。
作業答案
多分類問題
ex3.m
%% Machine Learning Online Class - Exercise 3 | Part 1: One-vs-all
% Instructions
% ------------
%
% This file contains code that helps you get started on the
% linear exercise. You will need to complete the following functions
% in this exericse:
%
% lrCostFunction.m (logistic regression cost function)
% oneVsAll.m
% predictOneVsAll.m
% predict.m
%
% For this exercise, you will not need to change any code in this file,
% or any other files other than those mentioned above.
%
%% Initialization
clear ; close all; clc
%% Setup the parameters you will use for this part of the exercise
input_layer_size = 400; % 20x20 Input Images of Digits
num_labels = 10; % 10 labels, from 1 to 10
% (note that we have mapped "0" to label 10)
%% =========== Part 1: Loading and Visualizing Data =============
% We start the exercise by first loading and visualizing the dataset.
% You will be working with a dataset that contains handwritten digits.
%
% Load Training Data
fprintf('Loading and Visualizing Data ...\n')
load('ex3data1.mat'); % training data stored in arrays X, y
m = size(X, 1);
% Randomly select 100 data points to display
rand_indices = randperm(m);
sel = X(rand_indices(1:100), :);
displayData(sel);
fprintf('Program paused. Press enter to continue.\n');
pause;
%% ============ Part 2a: Vectorize Logistic Regression ============
% In this part of the exercise, you will reuse your logistic regression
% code from the last exercise. You task here is to make sure that your
% regularized logistic regression implementation is vectorized. After
% that, you will implement one-vs-all classification for the handwritten
% digit dataset.
%
% Test case for lrCostFunction
fprintf('\nTesting lrCostFunction() with regularization');
theta_t = [-2; -1; 1; 2];
X_t = [ones(5,1) reshape(1:15,5,3)/10];
y_t = ([1;0;1;0;1] >= 0.5);
lambda_t = 3;
[J grad] = lrCostFunction(theta_t, X_t, y_t, lambda_t);
fprintf('\nCost: %f\n', J);
fprintf('Expected cost: 2.534819\n');
fprintf('Gradients:\n');
fprintf(' %f \n', grad);
fprintf('Expected gradients:\n');
fprintf(' 0.146561\n -0.548558\n 0.724722\n 1.398003\n');
fprintf('Program paused. Press enter to continue.\n');
pause;
%% ============ Part 2b: One-vs-All Training ============
fprintf('\nTraining One-vs-All Logistic Regression...\n')
lambda = 0.1;
[all_theta] = oneVsAll(X, y, num_labels, lambda);
fprintf('Program paused. Press enter to continue.\n');
pause;
%% ================ Part 3: Predict for One-Vs-All ================
pred = predictOneVsAll(all_theta, X);
fprintf('\nTraining Set Accuracy: %f\n', mean(double(pred == y)) * 100);
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)
%
J=1/m*(-y'*log(sigmoid(X*theta))-(1-y)'*log(1-sigmoid(X*theta)))+lambda*(theta'*theta-theta(1)^2)/2/m;
grad=X'*(sigmoid(X*theta)-y)/m;
temp=theta*lambda/m;
temp(1)=0;
grad=grad+temp;
% =============================================================
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
for i=1:num_labels
[all_theta(i,:)] = ...
fmincg (@(t)(lrCostFunction(t, X, y==i, 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.
%
for i=1:m
temp=all_theta*X(i,:)';
p(i)=find(temp==max(temp));
end
% =========================================================================
end