1. MURI Progress Report, June 2001 Advances in Approximate and Hybrid Reasoning for Decision Making Under Uncertainty Rina Dechter UC- Irvine Collaborators: Kalev Kask, Javier Larrosa, David Larkin, Robert Mateescu

2. MURI Progress Report, June 2001 Summary of Results Mini-clustering: a universal anytime approximation scheme. Applied to probabilistic inference and to Optimization, decision making tasks Hybrid processing of beliefs and constraints REES: Reasoning Engine Evaluation Shell. Online algorithms (S. Irani)

3. MURI Progress Report, June 2001 Outline Mini-clustering approximation; approximation by partitioning, a universal anytime scheme Applied to probabilistic inference Applied to Decision Optimization tasks Hybrid processing of beliefs and constraints REES: Reasoning Engine Evaluation Shell. Online algorithms (S. Irani)

4. MURI Progress Report, June 2001 Mini-Clustering : Approximation by partitioning Past work: Mini-bucket approximation for variable elimination Applied to optimization Used for static heuristic generation for search Experiments with coding tasks, medical diagnosis Progress this year Mini-clustering approximation of tree-clustering Applied to Belief updating Applied to optimization and search

5. MURI Progress Report, June 2001 Motivation Decision-making algorithms are all too complex (NP-Hard). The main bottleneck is probabilistic inference: determining the posterior beliefs given evidence to help forming the right decision. Consequently, approximate, anytime methods are essential to assist in advise-giving for decision making.

6. MURI Progress Report, June 2001 Automated reasoning Tasks

7. MURI Progress Report, June 2001 A Reasoning problem Graph A B D F C G zBelief updating: y  y =  X-y  j P j zMPE: y   = max X  j P j zCSP: y   =  X  j C j zMax-CSP: z   = min X  j F j

8. MURI Progress Report, June 2001 2 4 3 1 G E F C D B A Tree Decomposition

9. MURI Progress Report, June 2001 ABC 2 4 1 3 BEF EFG EF BF BC BCDF G E F C D B A Cluster Tree Elimination (join-tree clustering)

10. MURI Progress Report, June 2001 Time complexity: Exponential in the induced-width O (N  d w*+1 ) Space complexity: Exponential in the separator O ( N  d sep ) Tree clustering Complexity

11. MURI Progress Report, June 2001 Idea of Mini-clustering Reduce the exponent (i.e. size of the cluster); partition into mini-clusters. Accuracy-control parameter z = maximum number of variables in a mini-cluster The idea was explored for variable elimination (Mini-Bucket)

12. MURI Progress Report, June 2001 Idea of Mini-clustering Split a cluster into mini-clusters =>bound complexity

13. MURI Progress Report, June 2001 ABC 2 4 3 BEF EFG EF BF BC BCDF MC(3) algorithm - example

14. MURI Progress Report, June 2001 2 4 1 3 EF BF BC 2 4 1 3 EF BF BC Tree-clustering vs Mini-clustering

15. MURI Progress Report, June 2001 Properties of MC(z) MC(z) computes a bound on the joint probability P(X,e) of each variable and each of its values. Time & space complexity: O(n  hw*  exp(z)) Lower, Upper bounds and Mean approximations Approximation improves with z but takes more time

16. MURI Progress Report, June 2001 Experiments Algorithms: Exact IBP Gibbs sampling (GS) Mini-Clustering (MC(z)) Networks: Probabilistic Decoding networks Medical diagnosis: CPCS 54 Random noisy-OR networks Random networks

17. MURI Progress Report, June 2001 Performance on CPCS54 w*=15

18. MURI Progress Report, June 2001 N=50, P=2, w*=10 Noisy-OR Networks 1

19. MURI Progress Report, June 2001 N=50, P=3, w*=16 Random Networks 2

20. MURI Progress Report, June 2001 N=100, P=4, w*=11 Coding networks

21. MURI Progress Report, June 2001 Outline Mini-clustering approximation; approximation by partitioning, a universal anytime scheme Applied to probabilistic inference Applied to Optimization and decision-making tasks Hybrid processing of beliefs and constraints REES: Reasoning Engine Evaluation Shell. Online algorithms (S. Irani)

22. MURI Progress Report, June 2001 Constraint Optimization for Decision-making (COP) Global optimization: Find the best cost assignment subject to constraints Singleton optimality: Find the best cost-extension for every singleton variable-value assignment (X,a).

23. MURI Progress Report, June 2001 5 2 1 3 4 6 3 1 26 5 2 1 6 4 3 5 (a) (b) (c) 4 Example : COP C ij = X i  X j Tree-width = 3 sep(5,6) = {1, 5}

24. MURI Progress Report, June 2001 From Mini-bucket elimination to Mini-Bucket Tree Elimination 5 2 1 3 4 6 5 2 1 3 4 6

25. MURI Progress Report, June 2001 Branch and Bound with lower bound Heuristics BBMB(z), the earlier algorithm: Heuristic, computed by MB(z), is static, variable ordering fixed. BBBT(z), the new algorithm: Lower bound is computed at each node of the search by MC(z). Used for dynamic variable and value ordering.

26. MURI Progress Report, June 2001 Accuracy of MCTE(z)

27. MURI Progress Report, June 2001 BBBT(z) vs. BBMB(z) BBBT(z) vs BBMB(z), N=50

28. MURI Progress Report, June 2001 BBBT(z) vs. BBMB(z). BBBT(z) vs BBMB(z), N=100

29. MURI Progress Report, June 2001 Conclusion Mini-clustering, MC(z) extends partition-based approximation from mini-buckets to tree decompositions. For Probabilistic inference: For Optimization and decision-making tasks Empirical evaluation demonstrates its effectiveness and superiority (for certain types of problems).

30. MURI Progress Report, June 2001 Outline Mini-clustering approximation; approximation by partitioning, a universal anytime scheme Applied to probabilistic inference Applied to Optimization and decision tasks Processing beliefs and constraints REES: Reasoning Engine Evaluation Shell. Online algorithms (S. Irani)

31. MURI Progress Report, June 2001 Task A: Representation and Integration of Uncertain Information Challenges: Coherent and efficient extension of Bayesian networks to accommodate diverse types of information. Subtasks: Constraint-based information Temporal information Incomplete information

32. MURI Progress Report, June 2001 Motivation Complex queries for war scenarios: What is the probability that either plan1 or plan2 hit the target, when plan2 or plan 3 can divert enemy fire, under bad weather or poor communication. Observing that the enemy fire is coming either from direction 1 or direction 2, when direction 1 implies ground fire, what is the likelihood of being hit.

33. MURI Progress Report, June 2001 Hybrid Processing Beliefs and Constraints Hybrid deterministic and probabilistic Information Complex queries: Complex evidence structure All reduce to propositional queries over a Belief network.

34. MURI Progress Report, June 2001 Hybrid (continued) Deterministic queries and information can be handled as Conditional Probability Tables (CPTs) Drawbacks: computational properties such as constraint propagation and unit resolution are not exploited. Target: to exploit constraint processing whenever possible

35. MURI Progress Report, June 2001 A Hybrid Belief Network D G A B C F Belief network P(g,f,d,c,b,a) =P(g|f,d)P(f|c,b)P(d|b,a)P(b|a)P(c|a)P(a) Bucket G: P(G|F,D) Bucket F: P(F|B,C) Bucket D: P(D|A,B) Bucket C: P(C|A) Bucket B: P(B|A) Bucket A: P(A)

36. MURI Progress Report, June 2001 Bucket G: P(G|F,D) Bucket F: P(F|B,C) Bucket D: P(D|A,B) Bucket C: P(C|A) Bucket B: P(B|A) Bucket A: P(A) (a) regular Elim-CPE Bucket G: P(G|F,D) Bucket F: P(F|B,C) Bucket D: P(D|A,B) Bucket C: P(C|A) Bucket B: P(B|A) Bucket A: P(A) (b) Elim-CPE-D with clause extraction Variable elimination for a hybrid network:

37. MURI Progress Report, June 2001 Empirical evaluation Elim-CPE Elim-Hidden model clauses as CPT with hidden variables Elim-CPE-D extracts clauses from deterministic CPT’s Benchmarks: Insurance and Hailfinder networks Random networks

38. MURI Progress Report, June 2001 test instances of the insurance network with query parameters Insurance Network

39. MURI Progress Report, June 2001 48 test instances with network parameters and query parameters Elim-CPE vs. Elim-CPE-D

40. MURI Progress Report, June 2001 50 test instances, network parameters of and query parameters Averages over 35 test instances, network parameters of and query parameters Elim-CPE vs. Elim-Hidden

41. MURI Progress Report, June 2001 Averages of 50 instances with network parameters and varied number of evidence. Elim-CPE vs. Elim-D

42. MURI Progress Report, June 2001 Conclusion Elim-CPE: an extended variable elimination algorithm exploiting both constraints and probabilities Empirical evaluation demonstrate Elim-CPE highly more effective than regular algorithms (Elim-Hidden) Elim-CPE-D, extracting deterministic information from BN, improves performance and becomes more significant as deterministic information grows.

43. MURI Progress Report, June 2001 Outline Mini-clustering approximation; approximation by partitioning, a universal anytime scheme Applied to probabilistic inference Applied to Optimization and decision tasks Processing beliefs and constraints REES: Reasoning Engine Evaluation Shell. Online algorithms (S. Irani)

44. MURI Progress Report, June 2001 REES: Reasoning Engine Evaluation Shell Generalizable and Customizable : Consistent handling of reasoning tasks Handles manually and randomly generated problems with same user interface Add your own network types Use your own calculating engine Not limited by present AI problem types Created by Kyle Bolen and Kalev Kask Under direction of Dr. Rina Dechter

45. MURI Progress Report, June 2001 Interface Allows For: Easy parameter entry Quick access to choices Simple selection process

46. MURI Progress Report, June 2001 Customize To: Include only what you need Output to a file Run multiple instances Run multiple algorithms

47. MURI Progress Report, June 2001 Understand The Results Easily compare different algorithms View only the output you want

48. MURI Progress Report, June 2001 Outline Mini-clustering approximation; approximation by partitioning, a universal anytime scheme Applied to probabilistic inference Applied to Optimization and decision tasks Processing beliefs and constraints REES: Reasoning Engine Evaluation Shell. Online algorithms (S. Irani)

49. MURI Progress Report, June 2001 Online Load Balancing with Multiple Resources, S. Irani Tasks arrive in time and must be assigned to a server/agent as they arrive Each task requires a known amount of each resource. Goal is to make assignments so that all resources are evenly balanced among agents Results Online algorithm whose performance within 2r of optimal. (r = number of resources)

50. MURI Progress Report, June 2001 Dynamic Vehicle Routing Requests for service arrive at specific locations over a given area. Each request has a deadline A single server travels between location servicing requests Plan route of vehicle to maximize number of requests satisfied by deadline. Progress report for Sandy Irani

51. MURI Progress Report, June 2001 Dynamic Vehicle Routing Results: Two different online algorithms developed whose performance is provably close to optimal. (Which is better depends on parameters of the system) Lower bounds showing algorithms within a constant of best online algorithms. Progress report for Sandy Irani

52. MURI Progress Report, June 2001 Summary Mini-clustering approximation; approximation by partitioning, a universal anytime scheme Applied to probabilistic inference Applied to Optimization and decision tasks Processing beliefs and constraints REES: Reasoning Engine Evaluation Shell. Online algorithms (S. Irani)