深度学习 Deep Learning UFLDL 最新 Tutorial 学习笔记 1：Linear Regression

1 前言

Andrew Ng的UFLDL在2014年9月底更新了！

对于开始研究Deep Learning的童鞋们来说这真的是极大的好消息!

新的Tutorial相比旧的Tutorial增加了Convolutional Neural Network的内容，了解的童鞋都知道CNN在Computer Vision的重大影响。并且从新编排了内容及exercises。

新的UFLDL网址为：

http://ufldl.stanford.edu/tutorial/

2 Linear Regression 理论简述

对于线性回归Linear Regression，恐怕大部分童鞋都了解，简单的说

线性回归问题就是一个目标值y取决于一组输入值x，我们要寻找一个最合适的假设Hypothesis来描述这个y与x的关系，然后利用这个Hypothesis来预测新的输入x对应的y。

这是个简单的最优化问题，我们需要一个代价函数cost function来描述在training set样本中的y与通过h函数预测的y之间的差距，从而利用这个cost function通过Gradient Decent梯度下降法来计算h的最优参数从而得到最优的h。

因为是通过样本让计算机“学习”合适的参数theta，因此这是一个最基本的机器学习算法。

cost function：

J(θ)=12∑i(hθ(x(i))−y(i))2=12∑i(θ⊤x(i)−y(i))2

对theta做偏导：

Differentiating the cost function J(θ) as given above with respect to a particular parameter θj gives us:

∂J(θ)∂θj=∑ix(i)j(hθ(x(i))−y(i))

3 Linear Regression 练习

3.1 ex1a_linreg.m 分析

%%This exercise uses a data from the UCI repository:% Bache, K. & Lichman, M. (2013). UCI Machine Learning Repository% http://archive.ics.uci.edu/ml% Irvine, CA: University of California, School of Information and Computer Science.%%Data created by:% Harrison, D. and Rubinfeld, D.L.% ''Hedonic prices and the demand for clean air''% J. Environ. Economics & Management, vol.5, 81-102, 1978.%addpath ../commonaddpath ../common/minFunc_2012/minFuncaddpath ../common/minFunc_2012/minFunc/compiled% Load housing data from file.data = load('housing.data');  % housing data  506x14 data=data'; % put examples in columns  14x506  一般这里将每一个样本放在每一列% Include a row of 1s as an additional intercept feature.data = [ ones(1,size(data,2)); data ];  % 15x506    增加intercept term % Shuffle examples. 乱序 目的在于之后能够随机选取training set和test setsdata = data(:, randperm(size(data,2))); %randperm(n)用于随机生成1到n的排列% Split into train and test sets% The last row of 'data' is the median home price.train.X = data(1:end-1,1:400);   %选择前400个样本来训练，后面的样本来做测试train.y = data(end,1:400);test.X = data(1:end-1,401:end);test.y = data(end,401:end);m=size(train.X,2);  %训练样本数量n=size(train.X,1);  %每个样本的变量个数% Initialize the coefficient vector theta to random values.theta = rand(n,1); %随机生成初始theta 每个值在(0,1)之间% Run the minFunc optimizer with linear_regression.m as the objective.%% TODO:  Implement the linear regression objective and gradient computations% in linear_regression.m%tic; %Start a stopwatch timer. 开始计时options = struct('MaxIter', 200);theta = minFunc(@linear_regression, theta, options, train.X, train.y);fprintf('Optimization took %f seconds.\n', toc); %toc Read the stopwatch timer% Run minFunc with linear_regression_vec.m as the objective.%% TODO:  Implement linear regression in linear_regression_vec.m% using MATLAB's vectorization features to speed up your code.% Compare the running time for your linear_regression.m and% linear_regression_vec.m implementations.%% Uncomment the lines below to run your vectorized code.%Re-initialize parameters%theta = rand(n,1);%tic;%theta = minFunc(@linear_regression_vec, theta, options, train.X, train.y);%fprintf('Optimization took %f seconds.\n', toc);% Plot predicted prices and actual prices from training set.actual_prices = train.y;predicted_prices = theta'*train.X;% Print out root-mean-squared (RMS) training error.平方根误差train_rms=sqrt(mean((predicted_prices - actual_prices).^2));fprintf('RMS training error: %f\n', train_rms);% Print out test RMS erroractual_prices = test.y;predicted_prices = theta'*test.X;test_rms=sqrt(mean((predicted_prices - actual_prices).^2));fprintf('RMS testing error: %f\n', test_rms);% Plot predictions on test data.plot_prices=true;if (plot_prices)  [actual_prices,I] = sort(actual_prices); %从小到大排序价格，I为index  predicted_prices=predicted_prices(I);  plot(actual_prices, 'rx');  hold on;  plot(predicted_prices,'bx');  legend('Actual Price', 'Predicted Price');  xlabel('House #');  ylabel('House price ($1000s)');end

3.2 linear_regression.m code

function [f,g] = linear_regression(theta, X,y)  %  % Arguments:  %   theta - A vector containing the parameter values to optimize.  %   X - The examples stored in a matrix.  %       X(i,j) is the i'th coordinate of the j'th example.  %   y - The target value for each example.  y(j) is the target for example j.  %    m=size(X,2);  n=size(X,1);  f=0;  g=zeros(size(theta));  %  % TODO:  Compute the linear regression objective by looping over the examples in X.  %        Store the objective function value in 'f'.  %  % TODO:  Compute the gradient of the objective with respect to theta by looping over  %        the examples in X and adding up the gradient for each example.  Store the  %        computed gradient in 'g'.  %%% YOUR CODE HERE %%%% Step 1 : Compute f cost functionfor i = 1:m    f = f + (theta' * X(:,i) - y(i))^2;endf = 1/2*f;% Step 2: Compute gradient for j = 1:n    for i = 1:m        g(j) = g(j) + X(j,i)*(theta' * X(:,i) - y(i));    end    end

3.3 Result

Optimization took 3.374166 seconds.
RMS training error: 4.679871
RMS testing error: 4.865463

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