MACHINE LEARNING. Support Vector Machines. Alessandro Moschitti

Transcription

1 MACHINE LEARNING Support Vector Machines Alessandro Moschitti Department of information and communication technology University of Trento

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3 Summary Support Vector Machines Hard-margin SVMs Soft-margin SVMs

4 Communications No lecture tomorrow (neither Dec. 8) ML Exams 12 January 2011 at 9:00, 26 January 2011 at 9:00 Exercise in Lab A201 (Polo scientifico e tecnologico) Wednesday 15 and 22 December, 2011 Time:

5 Which hyperplane choose?

6 Classifier with a Maximum Margin Var 1 IDEA 1: Select the hyperplane with maximum margin Margin Margin Var 2

7 Support Vector Var 1 Support Vectors Margin Var 2

8 Support Vector Machine Classifiers Var 1 The margin is equal to 2 k w w w x + b = k w x + b = k k k w x + b = 0 Var 2

9 Support Vector Machines Var 1 The margin is equal to We need to solve 2 k w w w x + b = k k k w x w x + b = k Var 2 + b = 0 max 2 k w w x + b +k, if x is positive w x + b k, if x is negative

10 Support Vector Machines Var 1 There is a scale for which k=1. The problem transforms in: w w x + b = w x w x + b = 1 Var 2 + b = 0 max 2 w w x + b +1, if x is positive w x + b 1, if x is negative

11 Final Formulation max 2 w w x i + b +1, y i =1 w x i + b 1, y i = -1 max 2 w y i ( w x i + b) 1 min w 2 y i ( w x i + b) 1 min w 2 2 y i ( w x i + b) 1

12 Optimization Problem Optimal Hyperplane: Minimize Subject to τ ( w ) = 1 2 w 2 y i (( w x i ) + b) 1,i = 1,...,m The dual problem is simpler

13 Lagrangian Definition

14 Dual Optimization Problem

15 Dual Transformation Given the Lagrangian associated with our problem To solve the dual problem we need to evaluate: Let us impose the derivatives to 0, with respect to w

16 Dual Transformation (cont d) and wrt b Then we substituted them in the Lagrange function

17 Final Dual Problem

18 Khun-Tucker Theorem Necessary and sufficient conditions to optimality

19 Properties coming from constraints Lagrange constraints: Karush-Kuhn-Tucker constraints m a i y i = 0 i=1 w = m i=1 α i y i x i α i [y i ( x i w + b) 1] = 0, i = 1,...,m α i Support Vectors have not null To evaluate b, we can apply the following equation

20 Warning! On the graphical examples, we always consider normalized hyperplane (hyperplanes with normalized gradient) b in this case is exactly the distance of the hyperplane from the origin So if we have an equation not normalized we may have x w '+b = 0 with x = x,y ( ) and w '= 1,1 ( ) and b is not the distance

21 Warning! Let us consider a normalized gradient w = 1/ 2,1/ 2 ( ) ( x,y) ( 1/ 2,1/ 2) + b = 0 x / 2 + y/ 2 = b y = x b 2 Now we see that -b is exactly the distance. For x =0, we have the intersection with w distance projected on is -b b 2. This

22 Soft Margin SVMs Var 1 ξ i ξ i slack variables are added w w x + b = 1 Some errors are allowed but they should penalize the objective function w x + b = w x + b = 0 Var 2

23 Soft Margin SVMs Var 1 ξ i The new constraints are y i ( w x i + b) 1 ξ i x i where ξ i 0 w w x + b = w x w x + b = 1 Var 2 + b = 0 The objective function penalizes the incorrect classified examples min 1 2 w 2 +C ξ i i C is the trade-off between margin and the error

24 Dual formulation By deriving wrt w, ξ and b

25 Partial Derivatives

26 Substitution in the objective function δ ij of Kronecker

27 Final dual optimization problem

28 Soft Margin Support Vector Machines The algorithm tries to keep ξ i low and maximize the margin NB: The number of error is not directly minimized (NP-complete problem); the distances from the hyperplane are minimized If C, the solution tends to the one of the hard-margin algorithm Attention!!!: if C = 0 we get w = 0, since min 1 2 w 2 +C If C increases the number of error decreases. When C tends to infinite the number of errors must be 0, i.e. the hard-margin formulation ξ i i y i ( w x i + b) 1 ξ i x i ξ i 0 y i b 1 ξ i x i

29 Robusteness of Soft vs. Hard Margin SVMs Var 1 Var 1 ξ i ξ i w x + b = 0 Var 2 w x + b = 0 Var 2 Soft Margin SVM Hard Margin SVM

30 Soft vs Hard Margin SVMs Soft-Margin has ever a solution Soft-Margin is more robust to odd examples Hard-Margin does not require parameters

31 Parameters min 1 2 w 2 +C ξ i i = min 1 2 w 2 +C C ξ i ξ i i i C: trade-off parameter J: cost factor = min 1 2 w 2 +C J ξ + i i + ξ i i

32 Theoretical Justification

33 Definition of Training Set error Training Data f Empirical Risk (error) Risk (error) { } N : R ±1 R emp [ f ] = 1 m ( x 1,y 1 ),...,( x m,y m ) R N ±1 { } m i=1 1 2 f ( x i ) y i R[ f ] = 1 2 f ( x ) y dp( x,y)

34 Error Characterization (part 1) From PAC-learning Theory (Vapnik): R(α) R emp (α) + ϕ( d m, log(δ ) m ) ϕ( d, log(δ ) ) = d (log 2 m d +1) log(δ 4 ) m m m where d is thevc-dimension, m is the number of examples, δ is a bound on the probability to get such error and α is a classifier parameter.

35 There are many versions for different bounds

36 Error Characterization (part 2)

37 Ranking, Regression and Multiclassification

38 Sec The Ranking SVM [Herbrich et al. 1999, 2000; Joachims et al. 2002] The aim is to classify instance pairs as correctly ranked or incorrectly ranked This turns an ordinal regression problem back into a binary classification problem We want a ranking function f such that x i > x j iff f(x i ) > f(x j ) or at least one that tries to do this with minimal error Suppose that f is a linear function f(x i ) = w x i

39 The Ranking SVM Sec Ranking Model: f x i f (x i )

40 Sec The Ranking SVM Then (combining the two equations on the last slide): x i > x j iff w x i w x j > 0 x i > x j iff w (x i x j ) > 0 Let us then create a new instance space from such pairs: z k = x i x k y k = +1, 1 as x i, < x k

41 Support Vector Ranking 1 Given two examples we build one example (x i, x j )

42 Support Vector Regression (SVR) f(x) Solution: +ε 0 -ε Constraints: x

43 Support Vector Regression (SVR) f(x) ξ +ε 0 -ε Minimise: N 1 2 wt w + C ξ i + ξ i Constraints: ( * ) i=1 ξ* x

44 Support Vector Regression y i is not -1 or 1 anymore, now it is a value ε is the tollerance of our function value

45 From Binary to Multiclass classifiers Three different approaches: ONE-vs-ALL (OVA) Given the example sets, {E1, E2, E3, } for the categories: {C1, C2, C3, } the binary classifiers: {b1, b2, b3, } are built. For b1, E1 is the set of positives and E2 E3 is the set of negatives, and so on For testing: given a classification instance x, the category is the one associated with the maximum margin among all binary classifiers

46 From Binary to Multiclass classifiers ALL-vs-ALL (AVA) Given the examples: {E1, E2, E3, } for the categories {C1, C2, C3, } build the binary classifiers: {b1_2, b1_3,, b1_n, b2_3, b2_4,, b2_n,,bn-1_n} by learning on E1 (positives) and E2 (negatives), on E1 (positives) and E3 (negatives) and so on For testing: given an example x, all the votes of all classifiers are collected where b E1E2 = 1 means a vote for C1 and b E1E2 = -1 is a vote for C2 Select the category that gets more votes

47 From Binary to Multiclass classifiers Error Correcting Output Codes (ECOC) The training set is partitioned according to binary sequences (codes) associated with category sets. For example, indicates that the set of examples of C1,C3 and C5 are used to train the C classifier. The data of the other categories, i.e. C2 and C4 will be negative examples In testing: the code-classifiers are used to decode one the original class, e.g. C = 1 and C = 1 indicates that the instance belongs to C1 That is, the only one consistent with the codes

48 SVM-light: an implementation of SVMs Implements soft margin Contains the procedures for solving optimization problems Binary classifier Examples and descriptions in the web site: (

49 References A tutorial on Support Vector Machines for Pattern Recognition Downloadable article (Chriss Burges) The Vapnik-Chervonenkis Dimension and the Learning Capability of Neural Nets Downloadable Presentation Computational Learning Theory (Sally A Goldman Washington University St. Louis Missouri) Downloadable Article AN INTRODUCTION TO SUPPORT VECTOR MACHINES (and other kernel-based learning methods) N. Cristianini and J. Shawe-Taylor Cambridge University Press Check our library The Nature of Statistical Learning Theory Vladimir Naumovich Vapnik - Springer Verlag (December, 1999) Check our library

50 Exercise 1. The equations of SVMs for Classification, Ranking and Regression (you can get them from my slides). 2. The perceptron algorithm for Classification, Ranking and Regression (the last two you have to provide by looking at what you wrote in point (1)). 3. The same as point (2) by using kernels (write the kernel definition as introduction of this section).