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ER-DCOPS: A FRAMEWORK FOR DCOP WITH UNCERTAINTY IN CONSTRAINT - PowerPoint PPT Presentation

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1 ER-DCOPS: A FRAMEWORK FOR DCOP WITH UNCERTAINTY IN CONSTRAINT UTILITIES Tiep Le, Ferdinando Fioretto, William Yeoh, Tran Cao Son, Enrico Pontelli Computer Science Department New Mexico State University 2 OUTLINE BACKGROUND &

  1. 1 ER-DCOPS: A FRAMEWORK FOR DCOP WITH UNCERTAINTY IN CONSTRAINT UTILITIES Tiep Le, Ferdinando Fioretto, William Yeoh, Tran Cao Son, Enrico Pontelli Computer Science Department New Mexico State University

  2. 2 OUTLINE • BACKGROUND & MOTIVATION • ER-DCOP • ER-DPOP ALGORITHM • EXPERIMENTAL RESULTS • CONCLUSION

  3. 3 OUTLINE • BACKGROUND & MOTIVATION • ER-DCOP • ER-DPOP ALGORITHM • EXPERIMENTAL RESULTS • CONCLUSION

  4. 4 DISTRIBUTED CONSTRAINT OPTIMZATION PROBLEMS • DCOP P = < Χ , D, F, A, α > x 3 x 1 x 2

  5. 5 DISTRIBUTED CONSTRAINT OPTIMZATION PROBLEMS • DCOP P = < Χ , D, F, A, α > x 3 x 1 x 2 D 1 = D 2 = {0} D 3 = {0, 1}

  6. 6 DISTRIBUTED CONSTRAINT OPTIMZATION PROBLEMS • DCOP P = < Χ , D, F, A, α > x 3 f 13 f 23 x 1 x 2 D 1 = D 2 = {0} D 3 = {0, 1}

  7. 7 DISTRIBUTED CONSTRAINT OPTIMZATION PROBLEMS • DCOP P = < Χ , D, F, A, α > f 13 f 23 x 3 x 1 x 3 U 13 x 2 x 3 U 23 f 13 f 23 0 0 50 0 0 40 0 1 30 0 1 50 x 1 x 2 D 1 = D 2 = {0} D 3 = {0, 1}

  8. 8 DISTRIBUTED CONSTRAINT OPTIMZATION PROBLEMS • DCOP P = < Χ , D, F, A, α > f 13 f 23 x 3 x 1 x 3 U 13 x 2 x 3 U 23 f 13 f 23 0 0 50 0 0 40 0 1 30 0 1 50 x 1 x 2 D 1 = D 2 = {0} Worker x 1 owns variable x 1 Worker x 2 owns variable x 2 D 3 = {0, 1} Assistant robot x 3 owns variable x 3

  9. 9 DISTRIBUTED CONSTRAINT OPTIMZATION PROBLEMS • DCOP P = < Χ , D, F, A, α > • Goal: The assignment for all variables maximizes the aggregate utility f 13 f 23 x 3 x 1 x 3 U 13 x 2 x 3 U 23 f 13 f 23 0 0 50 0 0 40 0 1 30 0 1 50 x 1 x 2 D 1 = D 2 = {0} Worker x 1 owns variable x 1 Worker x 2 owns variable x 2 D 3 = {0, 1} Assistant robot x 3 owns variable x 3

  10. 10 MOTIVATION • In real-world applications, the utilities are stochastic. f 23 x 2 x 3 U 23 0 0 (Fail) 0 (Success) 40 0 1 (Fail) 0 (Success) 50

  11. 11 UR-DCOP • In real-world applications, the utilities are stochastic. f 23 x 2 x 3 U 23 Good Bad 0 0 (Fail) 0 50% 90% (Success) 40 50% 10% 0 1 (Fail) 0 90% 50% (Success) 50 10% 50% • Stochastic utilities can be sampled from a known probability distribution space.

  12. 12 MOTIVATION • In real-world applications, the utilities are stochastic. f 23 x 2 x 3 U 23 Good Bad 0 0 (Fail) 0 50% 90% (Success) 40 50% 10% 0 1 (Fail) 0 50% 20% (Success) 50 50% 80% • Stochastic utilities can be sampled from a known probability distribution space. • Expected-regret

  13. 13 MOTIVATION • In real-world applications, the utilities are stochastic. f 23 x 2 x 3 U 23 Good Bad 0 0 (Fail) 0 50% 90% (Success) 40 50% 10% 0 1 (Fail) 0 50% 20% (Success) 50 50% 80% • Stochastic utilities can be sampled from a known probability distribution space. ER-DCOP framework!

  14. 14 OUTLINE • BACKGROUND • ER-DCOP • ER-DPOP ALGORITHM • EXPERIMENTAL RESULTS • CONCLUSION

  15. 15 EXPECTED REGRET-DCOP (ER-DCOP) x 3 x 1 x 3 U 13 x 2 x 3 U 23 0 0 50 0 0 40 0 1 30 0 1 50 x 1 x 2

  16. 16 EXPECTED REGRET-DCOP (ER-DCOP) x 3 x 1 x 3 U 13 x 2 x 3 U 23 0 0 50 0 0 40 0 1 30 0 1 50 x 1 x 2 x 2 x 3 U 23 0 0 (Fail) 0 (Success) 40 0 1 (Fail) 0 (Success) 50

  17. 17 EXPECTED REGRET-DCOP (ER-DCOP) x 3 x 1 x 3 U 13 x 2 x 3 U 23 0 0 50 0 0 40 0 1 30 0 1 50 x 1 x 2 x 2 x 3 r 2 U 23 0 0 0 (Fail) 0 1 (Success) 40 0 1 0 (Fail) 0 1 (Success) 50

  18. 18 EXPECTED REGRET-DCOP (ER-DCOP) x 3 x 1 x 3 U 13 x 2 x 3 U 23 0 0 50 0 0 40 0 1 30 0 1 50 x 1 x 2 x 2 x 3 r 2 U 23 Good Bad 0 0 0 (Fail) 0 50% 90% Good: 12% 1 (Success) 40 50% 10% Bad: 88% 0 1 0 (Fail) 0 20% 50% 1 (Success) 50 80% 50%

  19. 19 EXPECTED REGRET-DCOP (ER-DCOP) • ER-DCOP P = < Χ , D, A, α , R, S, F> x 1 x 3 r 1 U 13 Good Bad x 2 x 3 r 2 U 23 Good Bad 0 0 0 (Fail) 0 10% 30% 0 0 0 (Fail) 0 50% 90% 1 (Success) 50 90% 70% 1 (Success) 40 50% 10% 0 1 0 (Fail) 0 30% 50% 0 1 0 (Fail) 0 20% 50% 1 (Success) 30 70% 50% 1 (Success) 50 80% 50%

  20. 20 EXPECTED REGRET-DCOP (ER-DCOP) • ER-DCOP P = < Χ , D, A, α , R, S, F> • belief of r 1 , belief of r 2 • x : joint belief for all random variables x 1 x 3 r 1 U 13 Good x 2 x 3 r 2 U 23 Good 0 0 0 (Fail) 0 10% 0 0 0 (Fail) 0 50% 1 (Success) 50 90% 1 (Success) 40 50% 0 1 0 (Fail) 0 30% 0 1 0 (Fail) 0 20% 1 (Success) 30 70% 1 (Success) 50 80%

  21. 21 EXPECTED REGRET-DCOP (ER-DCOP) • ER-DCOP P = < Χ , D, A, α , R, S, F> • Using Expected Utility (EU) consider only 1 joint belief of good weather x 1 x 3 r 1 U 13 Good EU x 2 x 3 r 2 U 23 Good EU 0 0 0 (Fail) 0 10% 45 0 0 0 (Fail) 0 50% 20 1 (Success) 50 90% 1 (Success) 40 50% 0 1 0 (Fail) 0 30% 21 0 1 0 (Fail) 0 20% 40 1 (Success) 30 70% 1 (Success) 50 80%

  22. 22 EXPECTED REGRET-DCOP (ER-DCOP) bad weather x 1 x 2 x 3 EU 0 0 0 39 0 0 1 40 Optimal assignment if bad weather (EU = 40) good weather x 1 x 2 x 3 EU Optimal assignment if good weather (EU = 65) 0 0 0 65 0 0 1 41 Belief Space Good: 12% Bad: 88%

  23. 23 EXPECTED REGRET-DCOP (ER-DCOP) bad weather x 1 x 2 x 3 EU Regret 0 0 0 39 40-39=1 0 0 1 40 40-40=0 Assignment x 1 = x 2 = x 3 = 0 has regret of 1 if bad weather good weather regret of 0 if good weather x 1 x 2 x 3 EU Regret 0 0 0 65 65-65=0 0 0 1 41 65-61=4 Belief Space Good: 12% Bad: 88%

  24. 24 EXPECTED REGRET-DCOP (ER-DCOP) bad weather x 1 x 2 x 3 EU Regret 0 0 0 39 40-39=1 88% 0 0 1 40 40-40=0 Expected-Regret (ER) x 1 x 2 x 3 ER 0 0 0 12%*0 + 88%*1 = 0.88 good weather 0 0 1 12%*4 + 88%*0 = 0.48 x 1 x 2 x 3 EU Regret 12% 0 0 0 65 65-65=0 0 0 1 41 65-61=4 Belief Space Good: 12% Bad: 88%

  25. 25 EXPECTED REGRET-DCOP (ER-DCOP) bad weather x 1 x 3 EU x 2 x 3 EU Regret 0 0 35 0 0 4 40-39=1 88% 0 1 15 0 1 25 40-40=0 Expected-Regret (ER) x 1 x 2 x 3 ER 0 0 0 12%*0 + 88%*1 = 0.88 good weather 0 0 1 12%*4 + 88%*0 = 0.48 x 1 x 3 EU x 2 x 3 EU Regret 12% 0 0 45 0 0 20 65-65=0 0 1 21 0 1 40 65-61=4 The solution minimizes the expected-regret

  26. 26 OUTLINE • BACKGROUND • ER-DCOP • ER-DPOP ALGORITHM • EXPERIMENTAL RESULTS • CONCLUSION

  27. 27 ER-DPOP • Phase 1: Generation of the pseudo-tree x 3 x 1 x 2

  28. 28 ER-DPOP • Phase 2: Resolution of subproblems x 1 x 3 r 1 U 13 Good Bad 0 0 0 (Fail) 0 10% 30% x 3 1 (Success) 50 90% 70% 0 1 0 (Fail) 0 30% 50% x 1 x 2 1 (Success) 30 70% 50% x 3 EU(Good) EU(Bad) 0 45 35 1 21 15 EU = Expected Utility

  29. 29 ER-DPOP • Phase 2: Resolution of subproblems x 2 x 3 r 2 U 23 Good Bad 0 0 0 (Fail) 0 50% 90% x 3 1 (Success) 40 50% 10% 0 1 0 (Fail) 0 20% 50% x 1 x 2 1 (Success) 50 80% 50% x 3 EU(Good) EU(Bad) 0 20 4 1 40 25 EU = Expected Utility

  30. 30 ER-DPOP • Phase 2: Resolution of subproblems x 3 EU(Good) EU(Bad) x 3 0 45+20=65 35+4=39 1 21+40=61 15+25=40 x 1 x 2 EU = Expected Utility

  31. 31 ER-DPOP • Phase 3: Resolution of the main problems • Generate DCOP with expected-regret as utilities • Use DPOP [Petcu et al. AAAI2007] to solve that DCOP x 1 x 3 EU(Good) EU(Bad) x 2 x 3 EU(Good) EU(Bad) 0 0 45 35 0 0 20 4 0 1 21 15 0 1 40 25 x 1 x 3 Expected-Regret x 2 x 3 Expected-Regret 0 0 12%*(45-45) + 0 0 12%*(20-20) + 88%*(15-35) = -17.6 88%*(25-4) = 18.48 0 1 12%*(45-21) + 0 1 12%*(20-40) + 88%*(15-15) = 2.88 88%*(25-25) = 2.4

  32. 32 ER-DPOP IMPLEMENTATIONS • GPU-ER-DPOP (GPU-based ER-DPOP) • Utilizes the parallelism offered by Graphical Processing Unit (GPU) to speed up computations in ER-DPOP • ASP-ER-DPOP (ASP-based ER-DPOP) • Prunes the search space offered by logic-programming based inference rules in Answer Set Programming (ASP)

  33. 33 RELATED WORK • UR-DCOP (F. Wu et al. AAAI 2014) • Beliefs of random variables are independent with values of decision variables; • Belief space does not exhibit probabilistic model; • Minimizing the worst-case loss (regret) over belief space.

  34. 34 OUTLINE • BACKGROUND • ER-DCOP • ER-DPOP ALGORITHM • EXPERIMENTAL RESULTS • CONCLUSION

  35. 35 EXPERIMENTAL RESULTS • Algorithms: • GPU-ER-DPOP • ASP-ER-DPOP • FRODO-ER (solve subproblems in Phase 2 sequentially) • Domains: • Random Graph (varying |X|, |D|, constraint density p 1 , constraint tightness p 2 , or belief space’s size) • Power Network Problems (varying Topology or |D|)

  36. 36 EXPERIMENTAL RESULTS |X| ASP-ER-DPOP GPU-ER-DPOP FRODO-ER 8 3.1 0.1 0.3 13 9.4 0.2 61.1 18 44.1 N/A N/A 23 120.8 N/A N/A runtime in second |D| ASP-ER-DPOP GPU-ER-DPOP FRODO-ER N/A: not available 4 4.5 0.1 1.8 6 8.9 0.1 33.6 8 22.2 1.2 143.2 10 80.4 4.8 N/A 12 121.2 15.4 N/A Random Graphs

  37. 37 EXPERIMENTAL RESULTS |A| = 13, |X| = 74, |F| = 51 |A| = 37, |X| = 218, |F| = 147 10 6 10 6 Simulated Runtime (ms) Simulated Runtime (ms) 10 5 10 5 10 4 10 4 10 3 10 3 ASP-ER ASP-ER Frodo-ER Frodo-ER 10 2 10 2 3 5 7 9 11 13 15 3 5 7 9 11 13 15 Domain Size Domain Size |A| = 124, |X| = 748, |F| = 497 10 6 Simulated Runtime (ms) 10 5 10 4 Power Network Problems 10 3 ASP-ER Frodo-ER 10 2 3 5 7 9 11 13 15 Domain Size

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