Linear Models for Regression CSE 4309 Machine Learning

Linear Models for Regression CSE 4309 – Machine Learning Vassilis Athitsos Computer Science and Engineering Department University of Texas at Arlington 1

The Regression Problem • 2

A Regression Example • circles: training data. • red curve: one possible solution: a line. • green curve: another possible solution: a cubic polynomial. 3

Linear Models for Regression • 4

Dot Product Version • 6

Common Choices for Basis Functions • 7

Common Choices for Basis Functions • 8

Common Choices for Basis Functions • 9

Common Choices for Basis Functions • 10

Common Choices for Basis Functions • 11

Global Versus Local Functions • 12

Linear Versus Nonlinear Functions • A linear function y(x, w) produces an output that depends linearly on both x and w. • Note that polynomial, Gaussian, and sigmoidal basis functions are nonlinear. • If we use nonlinear basis functions, then the regression process produces a function y(x, w) which is: – Linear to w. – Nonlinear to x. • It is important to remember: linear regression can be used to estimate nonlinear functions of x. – It is called linear regression because y is linear to w, NOT because y is linear to x. 13

Solving Regression Problems • There are different methods for solving regression problems. • We will study two approaches: – Least squares: find the weights w that minimize the squared error. – Regularized least squares: find the weights w that minimize the squared error, using some hand-picked regularization parameter λ. 14

The Gaussian Noise Assumption • 15

The Gaussian Noise Assumption • 16

Finding the Most Likely Solution • 17

Finding the Most Likely Solution • 18

Likelihood of the Training Data • 19

Likelihood of the Training Data • 20

Log Likelihood of the Training Data • 21

Log Likelihood of the Training Data 22

Log Likelihood of the Training Data • 23

Log Likelihood and Sum-of-Squares Error • 24

Minimizing Sum-of-Squares Error • 25

Minimizing Sum-of-Squares Error • 26

Minimizing Sum-of-Squares Error • 27

Minimizing Sum-of-Squares Error • 28

Minimizing Sum-of-Squares Error • 29

Minimizing Sum-of-Squares Error • 30

Minimizing Sum-of-Squares Error • 31

Minimizing Sum-of-Squares Error • 32

Minimizing Sum-of-Squares Error • 33

Minimizing Sum-of-Squares Error • 34

Minimizing Sum-of-Squares Error • 35

Minimizing Sum-of-Squares Error • 36

Minimizing Sum-of-Squares Error • 37

Minimizing Sum-of-Squares Error • Transposing both sides 38

Minimizing Sum-of-Squares Error • 39

Minimizing Sum-of-Squares Error • 40

Minimizing Sum-of-Squares Error • 41

Solving for β • 43

Least Squares Solution - Summary • 44

Numerical Issues • Black circles: training data. • Green curve: true function y(x)=sin(2πx). • Red curve: – Top figure: Fitted 7 -degree polynomial. – Bottom figure: Fitted 8 -degree polynomial. • Do you see anything strange? 45

Numerical Issues • The 8 -degree polynomial fits the data worse than the 7 degree polynomial. • Is this possible? 46

Numerical Issues • The 8 -degree polynomial fits the data worse than the 7 degree polynomial. • Mathematically, this is impossible. • 8 -degree polynomials are a superset of 7 -degree polynomials. • Thus, the best-fitting 8 -degree polynomial cannot fit the data worse than the best-fitting 7 degree polynomial. 47

Numerical Issues • 48

Numerical Issues • Work-around in Matlab: • inv(phi' * phi) leads to the incorrect 8 -degree polynomial fit below. • pinv(phi' * phi) leads to the correct 8 -degree polynomial fit above (almost identical to the 7 -degree result). 49

Sequential Learning • 50

Sequential Learning • 51

Sequential Learning • 52

Sequential Learning - Intuition • 53

Sequential Learning - Intuition • 54

Regularization • Here we have a training set of 10 points (shown as blue circles). • In green, we see the function that was used to generate these points (noise was added to that function). – It is a sinusoidal function. • In red we see the best-fitting polynomials of degree 1, 3, 9. • Interestingly, degree 3 matches the data well. – It would not match as well if we included points from more periods of the sinusoidal wave. 55

Regularization • The solution for degree 9 suffers severely from overfitting. 56

Regularization • We note that overfitting leads to very large magnitudes of parameters. Degree 1 Degree 3 Degree 9 w 0 0. 82 0. 31 0. 35 w 1 -1. 27 7. 99 232. 37 w 2 -25. 43 -5321. 83 w 3 17. 37 48568. 31 w 4 -231639. 30 w 5 640042. 26 w 6 -1061800. 52 w 7 1042400. 18 w 8 -557682. 99 w 9 125201. 43 57

Regularization • 58

Regularization w 0 0. 35 w 1 232. 37 4. 74 w 2 -5321. 83 -0. 77 w 3 48568. 31 -31. 97 w 4 -231639. 30 -3. 89 w 5 640042. 26 55. 28 w 6 -1061800. 52 41. 32 w 7 1042400. 18 -45. 95 w 8 -557682. 99 -91. 53 w 9 125201. 43 72. 68 A small λ solves the overfitting 59 problem in this case.

Regularized Least Squares • 60

Solving Regularized Least Squares • 61

Why Use Regularization? • 62

Impact of Parameter λ 63

Impact of Parameter λ 64

Impact of Parameter λ • As the previous figures show: – Low values of λ lead to polynomials whose values fluctuate more and more rapidly. • This can lead to increased overfitting. – High values of λ lead to flatter and flatter polynomials, that look more and more like straight lines. • This can lead to increased underfitting, or not fitting the data sufficiently. 65