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Summary of Results

Advances in Approximate and Hybrid Reasoning for Decision Making Under Uncertainty Rina Dechter UC- Irvine Collaborators: Kalev Kask, Javier Larrosa, David Larkin, Robert Mateescu. Summary of Results.

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Summary of Results

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  1. Advances in Approximate and Hybrid Reasoning for Decision Making Under UncertaintyRina DechterUC- IrvineCollaborators:Kalev Kask,Javier Larrosa,David Larkin,Robert Mateescu MURI Progress Report, June 2001

  2. 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) MURI Progress Report, June 2001

  3. 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) MURI Progress Report, June 2001

  4. 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 MURI Progress Report, June 2001

  5. 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. MURI Progress Report, June 2001

  6. Automated reasoning Tasks MURI Progress Report, June 2001

  7. A Reasoning problem Graph A • Belief updating: • y = X-y j Pj • MPE: •  = maxX j Pj • CSP: •  = X j Cj • Max-CSP: •  = minX j Fj C B F D G MURI Progress Report, June 2001

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

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

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

  11. 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) MURI Progress Report, June 2001

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

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

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

  15. 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 MURI Progress Report, June 2001

  16. 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 MURI Progress Report, June 2001

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

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

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

  20. 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) MURI Progress Report, June 2001

  21. 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). MURI Progress Report, June 2001

  22. 6 6 5 5 3 4 3 3 6 5 2 2 4 2 1 4 1 1 (c) (a) (b) Example : COP Cij= Xi Xj Tree-width = 3 sep(5,6) = {1, 5} MURI Progress Report, June 2001

  23. 6 6 3 5 3 5 2 2 4 4 1 1 From Mini-bucket elimination to Mini-Bucket Tree Elimination MURI Progress Report, June 2001

  24. 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. MURI Progress Report, June 2001

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

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

  27. 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). MURI Progress Report, June 2001

  28. 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) MURI Progress Report, June 2001

  29. 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 MURI Progress Report, June 2001

  30. 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. MURI Progress Report, June 2001

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

  32. 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 MURI Progress Report, June 2001

  33. A C B F D G A Hybrid Belief Network 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) 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) MURI Progress Report, June 2001

  34. Variable elimination for a hybrid network: 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) 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 (a) regular Elim-CPE MURI Progress Report, June 2001

  35. 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 MURI Progress Report, June 2001

  36. Insurance Network test instances of the insurance network with query parameters < 15, 5 > MURI Progress Report, June 2001

  37. Elim-CPE vs. Elim-CPE-D 48 test instances with network parameters < 80, 4, 75 > and query parameters < 0, 10 > MURI Progress Report, June 2001

  38. Elim-CPE vs. Elim-Hidden 50 test instances, network parameters of < 50, 5, 0 > and query parameters < 50, 15 > Averages over 35 test instances, network parameters of < 40, 5, 0 > and query parameters < 60, 10 > MURI Progress Report, June 2001

  39. 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. MURI Progress Report, June 2001

  40. 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) MURI Progress Report, June 2001

  41. REES: Reasoning Engine Evaluation Shell Created by Kyle Bolen and Kalev Kask Under direction of Dr. Rina Dechter • 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 MURI Progress Report, June 2001

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

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

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

  45. 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) MURI Progress Report, June 2001

  46. 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) MURI Progress Report, June 2001

  47. 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 MURI Progress Report, June 2001

  48. 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 MURI Progress Report, June 2001

  49. 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) MURI Progress Report, June 2001

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