Primal Dual Combinatorial Algorithms

1. Primal Dual Combinatorial Algorithms Qihui Zhu May 11, 2009

2. Outline • Packing and covering • Primal and dual problems • Online prediction using feedback • Primal-dual combinatorial algorithm • Applications and extensions

3. Packing Problem (0-1 Knapsack) • n objects with weights Wi and prices pi • A knapsack with capacity c • Pack objects maximizing the total price without exceeding the capacity

4. Packing Problem (0-1 Knapsack) • n objects with weights Wi and prices pi • A knapsack with capacity c • Pack objects maximizing the total price without exceeding the capacity Knapsack IP

5. Packing Problem (General) • n objects with weights Wijand prices pi • m constraints with capacity cjto fit • Pack objects maximizing the total price without exceeding the capacities Packing IP

6. Covering Problem • n objects to cover at least pi times • m sets with costs cj cover each covers Wji objects • Cover all objects with the minimal cost

7. Covering Problem • n objects to cover at least pi times • m sets with costs cj cover each covers Wji objects • Cover all objects with the minimal cost Covering IP

8. Fractional Packing Original form

9. Fractional Packing Original form Matrix form

10. Fractional Packing and Covering Are Dual Packing Covering

11. Fractional Packing and Covering Are Dual Packing Covering Covering: best upper bound on packing Packing: best lower bound on covering

12. Decision Version (PST 95) Given constraint set such that Feasibility For packing: Binary search for optimization

13. Flipped Sides of the Same Coin Think of the game of twenty questions... “Yes”certificate “No”certificate

14. Key Idea #1: Need Primal-Dual Algorithm Primal: Dual: “Yes” certificate “No” certificate

15. How to Generate the Certificates? Randomly guessing ...

16. How to Generate the Certificates? Primal and Dual need to communicate Randomly guessing ...

17. From Dual to Primal Efficient combinatorial algorithm

18. From Dual to Primal Oracle Given dual estimate and constraint set Let , find such that Efficient combinatorial algorithm

19. From Dual to Primal Oracle Given dual estimate and constraint set Let , find such that For packing, essentially ignore all capacity constraints Reduce to sorting over ! Complexity:

20. From Primal to Dual Combination of hyperplanes

21. From Primal to Dual Tilt the hyperplanes using feedback , ..., etc. One step not enough! Getting complicated over iterations... Online Prediction Combination of hyperplanes

22. Online Prediction Event Expert • Experts predicting some uncertain event

23. Gain some value from the world (adversarial) Value Online Prediction Event Expert • Experts predicting some uncertain event

24. Gain some value from the world (adversarial) • Linearly combine by weights Weight Value Online Prediction Event Expert • Experts predicting some uncertain event

25. Gain some value from the world (adversarial) Time • Linearly combine by weights • Long term value over time Weight Value Online Prediction Event Expert • Experts predicting some uncertain event

26. A Simplified Case • Only 2 experts • Value Combined history 1 0 1 0 1 1 0 Expert 1 ? ? ? ? ？ ? 0 1 0 1 0 0 1 ? ? ? ? ? ? Expert 2 Average playoff Regret Value

27. Strategy I Take the best from history? 1 0 1 0.5 Combined 0.9 0 0 0 0 0 0 0.9 0 1 0 1 1 0 Expert 1 1 1 0 1 0 1 0.1 1 0 1 0 0 1 0 1 0 1 0 1 Expert 2

28. Strategy II Linearly weighted by the cumulative values? 1 0 1 0.5 Combined 0 1 0 2/3 2/3 2/3 1/3 1 1 0 1 1 1 0 Expert 1 1 1 2/3 2/3 2/3 0 0 0 1 0 0 0 1 1/3 1 0 0 1/3 1/3 Expert 2

29. Strategy III Exponentially weighted by the cumulative values! 1 0 1 0.5 Combined 0 1 0 1 1 1 0 1 1 0 1 1 1 0 Expert 1 1 1 0 1-ε 1-ε 1-ε 0 0 1 0 0 0 1 ε 1 0 0 ε ε Expert 2

30. Key Idea #2: Need Multiplicative Feedback Primal: Dual: “Yes” certificate “No” certificate

31. From Primal to Dual: Continued Multiplicative Weight Update (MW) Let be the current state at step and be the “feedback” Combination of hyperplanes

32. Regret Bound Theorem(LW94) Suppose predictions of experts have value . Each time the predictions are combined by weights , where . Update weights using . After time

33. Regret Bound Theorem(LW94) Suppose predictions of experts have value . Each time the predictions are combined by weights , where . Update weights using . After time Proof.Consider the potential function Show that it is dominated by the best expert.

34. Primal-Dual Algorithms Primal Dual Oracle Multiplicative Weight Update

35. Algorithm Primal-Dual Combinatorial Algorithm Initialize: , Repeat Set Oracle If then Return infeasible, output Set Multiplicative Weight Update Until Return feasible solution

36. Analysis Primal-Dual Combinatorial Algorithm Oracle: if failed, infeasible Initialize: , Repeat Set Oracle If then Return infeasible, output Set Multiplicative Weight Update Until Return feasible solution

37. Analysis Oracle: if failed, infeasible Primal-Dual Combinatorial Algorithm Initialize: , Repeat Set Oracle If then Return infeasible, output Set Multiplicative Weight Update Until Return feasible solution Width: Regret bound: Feasible solution! Complexity depends on

38. Applications • Fractional packing and covering • Multicommodity flow • Held-Karp Bound for TSP • Semidefinite programming (SDP) relaxation • General convex programming • Boosting • Matrix game

39. Applications • Fractional packing and covering • Multicommodity flow • Held-Karp Bound for TSP • Semidefinite programming (SDP) relaxation • General convex programming • Boosting • Matrix game

40. Multicommodity Flow • Objective: maximizing total flow while respecting capacities

41. Multicommodity Flow • Objective: maximizing total flow while respecting capacities • Packing paths • Update: route some flow along shortest path Congestion Multicommodity Flow

42. Multicommodity Flow Primal-Dual Combinatorial Algorithm P : polytope of graph flows Initialize: , Repeat Set Oracle If then Return infeasible, output Set Multiplicative Weight Update Until Return feasible solution Oracle : shortest path keep pushing flows through Augment length (GK 98)

43. SDP Relaxation Max Cut (GW 95) Sparsest Cut (ARV 04) • Better bounds on Max Cut, Sparsest Cut, Max-2Sat, etc. • Finding good geometric embedding by SDP + rounding • Interior point method not scale- Primal SDP Dual SDP

44. Primal SDP Primal-Dual Combinatorial Algorithm Initialize: , Repeat Set Oracle If then Return infeasible, output Set Multiplicative Weight Update Until Return feasible solution P : set of semidefinite matrices Oracle : compute eigenvectors!

45. ! Dual SDP Primal-Dual Combinatorial Algorithm P & Oracle : same as packing Initialize: , Repeat Set Oracle If then Return infeasible, output Set Multiplicative Weight Update Until Return feasible solution Multiplicative Weight Update Use matrix exponential

46. Weighted Majority Algorithm/Boosting Primal-Dual Combinatorial Algorithm Initialize: , Repeat Set Oracle If then Return infeasible, output Set Multiplicative Weight Update Until Return feasible solution Events/Classification results: not controlled by us! Error on training examples

47. Summary Combinatorial subroutines

48. Summary Combinatorial subroutines Fast algorithms!

49. Conclusion • A computational paradigm • Applied to numerous problems • Cover at least convex problems • Fast approximation algorithms • Trade accuracy for time: O(1/ε) • Fast combinatorial algorithms for oracle (without solving LP!) • Flexibility for algorithm design • Multiplicative weight update: a principle way to use feedback • Free to separate and combine constraints • Width management is important • Most algorithms polynomial to width • Separate high width constraints: nested primal-dual • Decompose constraint set

50. Thank you!Questions …