Online Learning for Real-World Problems: A Framework for Binary Classification

1. Online Learning forReal-World Problems Koby Crammer University of Pennsylvania

2. Ofer Dekel Josheph Keshet Shai Shalev-Schwatz Yoram Singer Axel Bernal Steve Caroll Mark Dredze Kuzman Ganchev Ryan McDonald Artemis Hatzigeorgiu Fernando Pereira Fei Sha Partha Pratim Talukdar Thanks

3. Tutorial Context SVMs Real-World Data Online Learning Tutorial Multicass, Structured Optimization Theory

4. Online Learning Tyrannosaurus rex

5. Online Learning Triceratops

6. Online Learning Velocireptor Tyrannosaurus rex

7. Formal Setting – Binary Classification • Instances • Images, Sentences • Labels • Parse tree, Names • Prediction rule • Linear predictions rules • Loss • No. of mistakes

8. Predictions • Discrete Predictions: • Hard to optimize • Continuous predictions : • Label • Confidence

9. Loss Functions • Natural Loss: • Zero-One loss: • Real-valued-predictions loss: • Hinge loss: • Exponential loss (Boosting) • Log loss (Max Entropy, Boosting)

10. Loss Functions Hinge Loss Zero-One Loss 1 1

11. Online Framework • Initialize Classifier • Algorithm works in rounds • On round the online algorithm : • Receives an input instance • Outputs a prediction • Receives a feedback label • Computes loss • Updates the prediction rule • Goal : • Suffer small cumulative loss

12. Notation Abuse Linear Classifiers • Any Features • W.l.o.g. • Binary Classifiers of the form

13. Linear Classifiers (cntd.) • Prediction : • Confidence in prediction:

14. Margin • Margin of an example with respect to the classifier : • Note : • The set is separable iff there exists such that

15. Geometrical Interpretation

16. Geometrical Interpretation

17. Geometrical Interpretation

18. Geometrical Interpretation Margin <<0 Margin >0 Margin <0 Margin >>0

19. Hinge Loss

20. Separable Set

21. Inseparable Sets

22. Degree of Freedom - I The same geometrical hyperplane can be represented by many parameter vectors

23. Degree of Freedom - II Problem difficulty does not change if we shrink or expand the input space

24. Why Online Learning? • Fast • Memory efficient - process one example at a time • Simple to implement • Formal guarantees – Mistake bounds • Online to Batch conversions • No statistical assumptions • Adaptive • Not as good as a well designed batch algorithms

25. Update Rules • Online algorithms are based on an update rule which defines from (and possibly other information) • Linear Classifiers : find from based on the input • Some Update Rules : • Perceptron (Rosenblat) • ALMA (Gentile) • ROMMA (Li & Long) • NORMA (Kivinen et. al) • MIRA (Crammer & Singer) • EG (Littlestown and Warmuth) • Bregman Based (Warmuth)

26. Three Update Rules • The Perceptron Algorithm : • Agmon 1954; Rosenblatt 1952-1962, Block 1962, Novikoff 1962, Minsky & Papert 1969, Freund & Schapire 1999, Blum & Dunagan 2002 • Hildreth’s Algorithm : • Hildreth 1957 • Censor & Zenios 1997 • Herbster 2002 • Loss Scaled : • Crammer & Singer 2001, • Crammer & Singer 2002

27. The Perceptron Algorithm • If No-Mistake • Do nothing • If Mistake • Update • Margin after update :

28. Geometrical Interpretation

29. Relative Loss Bound • For any competitor prediction function • We bound the loss suffered by the algorithm with the loss suffered by Cumulative Loss Suffered by the Algorithm Sequence of Prediction Functions Cumulative Loss of Competitor

30. Relative Loss Bound • For any competitor prediction function • We bound the loss suffered by the algorithm with the loss suffered by Inequality Possibly Large Gap Regret Extra Loss Competitiveness Ratio

31. Relative Loss Bound • For any competitor prediction function • We bound the loss suffered by the algorithm with the loss suffered by Grows With T Grows With T Constant

32. Relative Loss Bound • For any competitor prediction function • We bound the loss suffered by the algorithm with the loss suffered by Best Prediction Function in hindsight for the data sequence

33. Remarks • If the input is inseparable, then the problem of finding a separating hyperplane which attains less then M errors is NP-hard (Open hemisphere) • Obtaining a zero-one loss bound with a unit competitiveness ratio is as hard as finding a constant approximating error for the Open Hemisphere problem. • Bound of the number of mistakes the perceptron makes with the hinge loss of any competitor

34. Definitions • Any Competitor • The parameters vector can be chosen using the input data • The parameterized hinge loss of on • True hinge loss • 1-norm and 2-norm of hinge loss

35. Geometrical Assumption • All examples are bounded in a ball of radius R

36. Perceptron’s Mistake Bound • Bounds : • If the sample is separable then

37. [FS99, SS05] Proof - Intuition • Two views : • The angle between and decreases with • The following sum is fixed as we make more mistakes, our solution is better

38. [C04] Proof • Define the potential : • Bound it’s cumulative sum from above and below

39. Proof • Bound from above : Telescopic Sum Zero Vector Non-Negative

40. Proof • Bound From Below : • No error on tth round • Error on tth round

41. Proof • We bound each term :

42. Proof • Bound From Below : • No error on tth round • Error on tth round • Cumulative bound :

43. Proof • Putting both bounds together : • We use first degree of freedom (and scale) : • Bound :

44. Proof • General Bound : • Choose : • Simple Bound : Objective of SVM

45. Proof • Better bound : optimize the value of

46. Remarks • Bound does not depend on dimension of the feature vector • The bound holds for all sequences. It is not tight for most real world data • But, there exists a setting for which it is tight

47. Three Bounds

48. Separable Case • Assume there exists such that for all examples Then all bounds are equivalent • Perceptron makes finite number of mistakes until convergence (not necessarily to )

49. Separable Case – Other Quantities • Use 1st (parameterization) degree of freedom • Scale the such that • Define • The bound becomes

50. Separable Case - Illustration