Discriminative, Unsupervised, Convex Learning

1. Discriminative, Unsupervised, Convex Learning Dale Schuurmans Department of Computing Science University of Alberta MITACS Workshop, August 26, 2005

2. Current Research Group PhD Tao Wang reinforcement learning PhD Ali Ghodsi dimensionality reduction PhD Dana Wilkinson action-based embedding PhD Yuhong Guo ensemble learning PhD Feng Jiao bioinformatics PhD Jiayuan Huang transduction on graphs PhD Qin Wang statistical natural language PhD Adam Milstein robotics, particle filtering PhD Dan Lizotte optimization, everything PhD Linli Xu unsupervised SVMs PDF Li Cheng computer vision

3. Current Research Group PhD Tao Wang reinforcement learning PhD Dana Wilkinson action-based embedding PhD Feng Jiao bioinformatics PhD Qin Wang statistical natural language PhD Dan Lizotte optimization, everything PDF Li Cheng computer vision

4. Today I will talk about: One Current Research Direction Learning Sequence Classifiers (HMMs) • Discriminative • Unsupervised • Convex EM?

5. Outline • Unsupervised SVMs • Discriminative, unsupervised, convex HMMs • Tao, Dana, Feng, Qin, Dan, Li

7. Unsupervised Support Vector Machines Joint work with Linli Xu

8. Main Idea • UnsupervisedSVMs (and semi-supervised SVMs) • Harder computational problem than SVMs • Convex relaxation – Semidefinite program (Polynomial time)

9. Background: Two-class SVM • Supervised classification learning • Labeled data  linear discriminant • Classification rule: + Some better than others?

10. Maximum Margin Linear Discriminant Choose a linear discriminant to maximize

11. Unsupervised Learning • Given unlabeled data, how to infer classifications? • Organize objects into groups — clustering

12. Idea: Maximum Margin Clustering • Given unlabeled data, find maximum margin separating hyperplane • Clusters the data • Constraint:class balance: bound difference in sizes between classes

13. Challenge • Find label assignment that results in a large margin • Hard • Convex relaxation – based on semidefinite programming

14. How to Derive Unsupervised SVM? Two-class case: • Start with Supervised Algorithm Given vector of assignments, y, solve Inv. sq. margin

15. How to Derive Unsupervised SVM? • Think of as a function of y Goal: Choose y to minimize inverse squared margin If given y, would then solve Problem: not a convex function of y Inv. sq. margin

16. How to Derive Unsupervised SVM? • Re-express problem with indicators comparing y labels New variables: An equivalence relation matrix If given y, would then solve Inv. sq. margin

17. How to Derive Unsupervised SVM? • Re-express problem with indicators comparing y labels New variables: An equivalence relation matrix If given M, would then solve Note:convex function of M Inv. sq. margin Maximum of linear functions is convex

18. How to Derive Unsupervised SVM? • Get constrained optimization problem Solve for M encodes an equivalence relation iff Not convex! Class balance 

19. How to Derive Unsupervised SVM? • Get constrained optimization problem Solve for M encodes an equivalence relation iff

20. How to Derive Unsupervised SVM? • Relax indicator variables to obtain a convex optimization problem Solve for M

21. How to Derive Unsupervised SVM? • Relax indicator variables to obtain a convex optimization problem Solve for M Semidefinite program

22. Multi-class Unsupervised SVM? • Start with Supervised Algorithm Given vector of assignments, y, solve Margin loss (Crammer & Singer 01)

23. Multi-class Unsupervised SVM? • Think of as a function of y Goal: Choose y to minimize margin loss If given y, would then solve Margin loss Problem: not a convex function of y (Crammer & Singer 01)

24. Multi-class Unsupervised SVM? • Re-express problem with indicators comparing y labels New variables: M & D If given y, would then solve Margin loss (Crammer & Singer 01)

25. Multi-class Unsupervised SVM? • Re-express problem with indicators comparing y labels New variables: M & D If given MandD, would then solve Margin loss convex function of M & D

26. Multi-class Unsupervised SVM? • Get constrained optimization problem Solve for MandD Class balance 

27. Multi-class Unsupervised SVM? • Relax indicator variables to obtain a convex optimization problem Solve for MandD

28. Multi-class Unsupervised SVM? • Relax indicator variables to obtain a convex optimization problem Solve for MandD Semidefinite program

29. Experimental Results SemiDef Spectral Clustering Kmeans

30. Experimental Results

31. Experimental Results Percentage of misclassification errors Digit dataset

32. Extension to Semi-Supervised Algorithm MatrixM :

33. Experimental Results Percentage of misclassification errors Face dataset

34. Experimental Results

36. Discriminative, Unsupervised, Convex HMMs Joint work with Linli Xu With help from Li Cheng and Tao Wang

37. Must coordinate local classifiers Hidden Markov Model • Joint probability model • Viterbi classifier “hidden” state observations

38. HMM Training: Supervised • Given Models input distribution Maximum likelihood Conditional likelihood Discriminative (CRFs)

39. Marginal likelihood HMM Training: Unsupervised • Given only • Now what? EM! Exactly the part we don’t care about

40. HMM Training: Unsupervised • Given only The problem with EM: • Not convex • Wrong objective • Too popular • Doesn’t work

41. HMM Training: Unsupervised • Given only The dream: • Convex training • Discriminative training When will someone invent unsupervised CRFs?

42. HMM Training: Unsupervised • Given only The question: • How to learn effectively without seeing any y’s?

43. HMM Training: Unsupervised • Given only The question: • How to learn effectively without seeing any y’s? The answer: • That’s what we already did!  Unsupervised SVMs

44. HMM Training: Unsupervised • Given only The plan: single sequence SVM  M3N supervised    unsupervised unsup SVM ?

45. M3N: Max Margin Markov Nets • Relational SVMs • Supervised training: • Given • Solve factored QP

46. Unsupervised M3Ns • Strategy • Start with supervised M3N QP • y-labels  re-express in local M,D equivalence relations • Impose class-balance • Relax non-convex constraints • Then solve a really big SDP • But still polynomial size

47. Unsupervised M3Ns • SDP

48. Some Initial Results • Synthetic HMM • Protein Secondary Structure pred.

50. Current Research Group PhD Tao Wang reinforcement learning PhD Dana Wilkinson action-based embedding PhD Feng Jiao bioinformatics PhD Qin Wang statistical natural language PhD Dan Lizotte optimization, everything PDF Li Cheng computer vision