Counterexample-guided Cartesian Abstraction Refinement and Saturated - - PowerPoint PPT Presentation

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Counterexample-guided Cartesian Abstraction Refinement and Saturated - - PowerPoint PPT Presentation

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Counterexample-guided Cartesian Abstraction Refinement and Saturated Cost Partitioning for Optimal Classical Planning Jendrik Seipp February 28, 2018 University of Basel Planning Find a sequence of actions that achieves a goal. 1/35 Optimal

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SLIDE 1

Counterexample-guided Cartesian Abstraction Refinement and Saturated Cost Partitioning for Optimal Classical Planning

Jendrik Seipp February 28, 2018

University of Basel

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SLIDE 2

Planning

Find a sequence of actions that achieves a goal.

1/35

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SLIDE 3

Optimal Classical Planning

drive drive drive load-in-A unload-in-B unload-in-A load-in-B

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SLIDE 4

Optimal Classical Planning: Example Abstraction

drive drive drive load-in-A unload-in-A unload-in-B load-in-B

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SLIDE 5

Abstraction Heuristics

  • abstraction heuristics never overestimate → admissible
  • A∗ + admissible heuristic → optimal plan
  • higher accuracy → better guidance
  • how to create abstractions?

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SLIDE 6

Abstraction Heuristics

  • abstraction heuristics never overestimate → admissible
  • A∗ + admissible heuristic → optimal plan
  • higher accuracy → better guidance
  • how to create abstractions?

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SLIDE 7

Counterexample-guided Cartesian Abstraction Refinement

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SLIDE 8

Counterexample-guided Abstraction Refinement (CEGAR)

CEGAR Algorithm

  • start with coarse abstraction
  • until finding concrete solution or running out of time:
  • find abstract solution
  • check if and why it fails in the real world
  • refine abstraction

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SLIDE 9

Example Refinement

drive, (un)load-in-A, (un)load-in-B

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SLIDE 10

Example Refinement

drive, (un)load-in-A drive unload-in-B load-in-B

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SLIDE 11

Example Refinement

drive drive drive load-in-A unload-in-A unload-in-B load-in-B

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SLIDE 12

Classes of Abstractions

Cartesian Abstractions

  • relation to other classes of abstractions?

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SLIDE 13

Projection (PDB)

drive drive drive load-in-A unload-in-A unload-in-B load-in-B

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SLIDE 14

Cartesian Abstraction

drive drive (un)load-in-A (un)load-in-B drive

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SLIDE 15

Merge-and-shrink Abstraction

(un)load-in-B drive drive drive (un)load-in-A

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SLIDE 16

Classes of Abstractions

  • Projections (PDBs)

refinement at least doubles number of states

  • Cartesian Abstractions

allow efficient and fine-grained refinement

  • Merge-and-shrink Abstractions

refinement complicated and expensive

9/35

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SLIDE 17

Solved Tasks

700 800 900 1,000 1,100 PhO-Sys2 M&S iPDB

737 808 881

700 800 900 1,000 1,100 CEGAR

706

10/35

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SLIDE 18

CEGAR Drawbacks

Diminishing Returns

  • finding solutions takes longer
  • heuristic values only increase logarithmically

multiple smaller abstractions

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SLIDE 19

CEGAR Drawbacks

Diminishing Returns

  • finding solutions takes longer
  • heuristic values only increase logarithmically

→ multiple smaller abstractions

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SLIDE 20

Task Decomposition by Goals

  • build abstraction for each goal fact
  • problem: tasks with single goal fact

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SLIDE 21

Task Decomposition by Goals

  • build abstraction for each goal fact
  • problem: tasks with single goal fact

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SLIDE 22

Task Decomposition by Landmarks

  • build abstraction for each fact landmark

drive drive drive load-in-A unload-in-B unload-in-A load-in-B

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SLIDE 23

Task Decomposition by Landmarks

  • build abstraction for each fact landmark

drive drive drive load-in-A unload-in-B unload-in-A load-in-B

13/35

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SLIDE 24

Multiple Heuristics

how to combine multiple heuristics?

s1,s2 s3 s4,s5 4 4 1 1 s1 s2,s3,s4 s5 4 4 1 1

h1 s2 5 h2 s2 4 maximize over estimates:

  • h s2

5

  • only selects best heuristic
  • does not combine heuristics

14/35

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SLIDE 25

Multiple Heuristics

how to combine multiple heuristics?

s1,s2 s3 s4,s5 4 4 1 1 s1 s2,s3,s4 s5 4 4 1 1

h1 s2 5 h2 s2 4 maximize over estimates:

  • h s2

5

  • only selects best heuristic
  • does not combine heuristics

14/35

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SLIDE 26

Multiple Heuristics

how to combine multiple heuristics?

s1,s2 s3 s4,s5 4 4 1 1 s1 s2,s3,s4 s5 4 4 1 1

h1(s2) = 5 h2(s2) = 4 maximize over estimates:

  • h s2

5

  • only selects best heuristic
  • does not combine heuristics

14/35

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SLIDE 27

Multiple Heuristics

how to combine multiple heuristics?

s1,s2 s3 s4,s5 4 4 1 1 s1 s2,s3,s4 s5 4 4 1 1

h1(s2) = 5 h2(s2) = 4 maximize over estimates:

  • h(s2) = 5
  • only selects best heuristic
  • does not combine heuristics

14/35

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SLIDE 28

Multiple Heuristics

how to combine multiple heuristics?

s1,s2 s3 s4,s5 4 4 1 1 s1 s2,s3,s4 s5 4 4 1 1

h1(s2) = 5 h2(s2) = 4 maximize over estimates:

  • h(s2) = 5
  • only selects best heuristic
  • does not combine heuristics

14/35

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SLIDE 29

Multiple Heuristics: Cost Partitioning

Cost Partitioning

  • split operator costs among heuristics
  • sum of costs must not exceed original cost

s1,s2 s3 s4,s5 4 4 1 1 s1 s2,s3,s4 s5 4 4 1 1

h s2 3 3 6

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SLIDE 30

Multiple Heuristics: Cost Partitioning

Cost Partitioning

  • split operator costs among heuristics
  • sum of costs must not exceed original cost

s1,s2 s3 s4,s5 2 1 1 s1 s2,s3,s4 s5 1 3 1

h s2 3 3 6

15/35

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SLIDE 31

Multiple Heuristics: Cost Partitioning

Cost Partitioning

  • split operator costs among heuristics
  • sum of costs must not exceed original cost

s1,s2 s3 s4,s5 2 1 1 s1 s2,s3,s4 s5 1 3 1

h(s2) = 3 + 3 = 6

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SLIDE 32

Saturated Cost Partitioning

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SLIDE 33

Saturated Cost Partitioning

Saturated Cost Partitioning Algorithm

  • order heuristics, then for each heuristic h:
  • use minimum costs preserving all estimates of h
  • use remaining costs for subsequent heuristics

s1,s2 s3 s4,s5 4 4 1 1 s1 s2,s3,s4 s5 4 4 1 1

h s2 5 3 8

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SLIDE 34

Saturated Cost Partitioning

Saturated Cost Partitioning Algorithm

  • order heuristics, then for each heuristic h:
  • use minimum costs preserving all estimates of h
  • use remaining costs for subsequent heuristics

s1,s2 s3 s4,s5 4 4 1 1 s1 s2,s3,s4 s5

h s2 5 3 8

16/35

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SLIDE 35

Saturated Cost Partitioning

Saturated Cost Partitioning Algorithm

  • order heuristics, then for each heuristic h:
  • use minimum costs preserving all estimates of h
  • use remaining costs for subsequent heuristics

s1,s2 s3 s4,s5 4 1 1 s1 s2,s3,s4 s5

h s2 5 3 8

16/35

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SLIDE 36

Saturated Cost Partitioning

Saturated Cost Partitioning Algorithm

  • order heuristics, then for each heuristic h:
  • use minimum costs preserving all estimates of h
  • use remaining costs for subsequent heuristics

s1,s2 s3 s4,s5 4 1 1 s1 s2,s3,s4 s5 3 1

h s2 5 3 8

16/35

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SLIDE 37

Saturated Cost Partitioning

Saturated Cost Partitioning Algorithm

  • order heuristics, then for each heuristic h:
  • use minimum costs preserving all estimates of h
  • use remaining costs for subsequent heuristics

s1,s2 s3 s4,s5 4 1 1 s1 s2,s3,s4 s5 3

h s2 5 3 8

16/35

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SLIDE 38

Saturated Cost Partitioning

Saturated Cost Partitioning Algorithm

  • order heuristics, then for each heuristic h:
  • use minimum costs preserving all estimates of h
  • use remaining costs for subsequent heuristics

s1,s2 s3 s4,s5 4 1 1 s1 s2,s3,s4 s5 3

h(s2) = 5 + 3 = 8

16/35

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SLIDE 39

Solved Tasks

700 800 900 1,000 1,100 PhO-Sys2 M&S iPDB

737 808 881

700 800 900 1,000 1,100 CEGAR

706

17/35

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SLIDE 40

Solved Tasks

700 800 900 1,000 1,100 PhO-Sys2 M&S iPDB

737 808 881

700 800 900 1,000 1,100 CEGAR goals

706 774

17/35

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SLIDE 41

Solved Tasks

700 800 900 1,000 1,100 PhO-Sys2 M&S iPDB

737 808 881

700 800 900 1,000 1,100 CEGAR goals landmarks

706 774 785

17/35

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SLIDE 42

Solved Tasks

700 800 900 1,000 1,100 PhO-Sys2 M&S iPDB

737 808 881

700 800 900 1,000 1,100 CEGAR goals landmarks LMs+goals

706 774 785 798

17/35

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SLIDE 43

Order of Heuristics Is Important

s1,s2 s3 s4,s5 4 4 1 1 s1 s2,s3,s4 s5 4 4 1 1

hSCP s2 5 3 8 hSCP s2 3 4 7

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SLIDE 44

Order of Heuristics Is Important

s1,s2 s3 s4,s5 4 1 1 s1 s2,s3,s4 s5 3

hSCP

→ (s2) = 5 + 3 = 8

hSCP s2 3 4 7

18/35

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SLIDE 45

Order of Heuristics Is Important

s1,s2 s3 s4,s5 3 s1 s2,s3,s4 s5 1 4 1

hSCP

→ (s2) = 5 + 3 = 8

hSCP

← (s2) = 3 + 4 = 7 18/35

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SLIDE 46

Finding a Good Order

  • n heuristics → n! orders

search for good order: greedy initial order + optimization

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SLIDE 47

Finding a Good Order

  • n heuristics → n! orders

→ search for good order: greedy initial order + optimization

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SLIDE 48

Greedy Orders

Goal: high estimates and low costs

  • rder by heuristic/costs ratio

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SLIDE 49

Greedy Orders

Goal: high estimates and low costs → order by heuristic/costs ratio

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SLIDE 50

Solved Tasks

700 800 900 1,000 1,100 PhO-Sys2 M&S iPDB

737 808 881

700 800 900 1,000 1,100 CEGAR goals landmarks LMs+goals

706 774 785 798

21/35

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SLIDE 51

Solved Tasks

700 800 900 1,000 1,100 PhO-Sys2 M&S iPDB

737 808 881

700 800 900 1,000 1,100 CEGAR goals landmarks LMs+goals greedy

706 774 785 798 866

21/35

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SLIDE 52

Optimized Orders

Optimization: finding initial order usually only first step Hill-climbing Search

  • start with initial order
  • until no better successor found:
  • switch positions of two heuristics
  • commit to first improving successor

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SLIDE 53

Solved Tasks

700 800 900 1,000 1,100 PhO-Sys2 M&S iPDB

737 808 881

700 800 900 1,000 1,100 CEGAR goals landmarks LMs+goals greedy

706 774 785 798 866

23/35

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SLIDE 54

Solved Tasks

700 800 900 1,000 1,100 PhO-Sys2 M&S iPDB

737 808 881

700 800 900 1,000 1,100 CEGAR goals landmarks LMs+goals greedy

  • ptimized

706 774 785 798 866 881

23/35

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SLIDE 55

One Order Is Not Enough

s1,s2 s3 s4,s5 4 4 1 1 s1 s2,s3,s4 s5 4 4 1 1

hSCP

→ (s2) = 8

hSCP

← (s2) = 7

hSCP

→ (s4) = 3

hSCP

← (s4) = 4 24/35

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SLIDE 56

Multiple Orders

Approach:

  • compute saturated cost partitioning for multiple orders
  • maximize over heuristic estimates

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SLIDE 57

Solved Tasks

700 800 900 1,000 1,100 PhO-Sys2 M&S iPDB

737 808 881

700 800 900 1,000 1,100 CEGAR goals landmarks LMs+goals greedy

  • ptimized

706 774 785 798 866 881

26/35

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SLIDE 58

Solved Tasks

700 800 900 1,000 1,100 PhO-Sys2 M&S iPDB

737 808 881

700 800 900 1,000 1,100 CEGAR goals landmarks LMs+goals greedy

  • ptimized

multiple

706 774 785 798 866 881 982

26/35

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SLIDE 59

Multiple Orders

Problems:

  • many useless orders
  • slow evaluation

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SLIDE 60

Diverse Orders

Diversification Algorithm

  • sample 1000 states
  • start with empty set of orders
  • until time limit is reached:
  • generate an optimized order
  • if a sample profits from it, keep it
  • otherwise, discard it

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SLIDE 61

Solved Tasks

700 800 900 1,000 1,100 PhO-Sys2 M&S iPDB

737 808 881

700 800 900 1,000 1,100 CEGAR goals landmarks LMs+goals greedy

  • ptimized

multiple

706 774 785 798 866 881 982

29/35

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SLIDE 62

Solved Tasks

700 800 900 1,000 1,100 PhO-Sys2 M&S iPDB

737 808 881

700 800 900 1,000 1,100 CEGAR goals landmarks LMs+goals greedy

  • ptimized

multiple diverse

706 774 785 798 866 881 982 994

29/35

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SLIDE 63

Comparison of Cost Partitioning Algorithms

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SLIDE 64

Theoretical Comparison

UCP Uniform Cost Partitioning distribute costs evenly among relevant heuristics

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SLIDE 65

Theoretical Comparison

GZOCP UCP Greedy Zero-one Cost Partitioning

  • rder heuristics and give full cost to first relevant heuristic

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SLIDE 66

Theoretical Comparison

GZOCP PhO UCP Post-hoc Optimization compute weight for each heuristic and return weighted sum

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SLIDE 67

Theoretical Comparison

GZOCP PhO CAN UCP Canonical Heuristic maximum over sums of independent heuristic subsets

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SLIDE 68

Theoretical Comparison

GZOCP PhO CAN UCP ≻ ≻ ≻ Pommerening et al. 2013

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SLIDE 69

Theoretical Comparison

GZOCP PhO CAN UCP ≻ ≻ ≻ ≻ ≻ ≻

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SLIDE 70

Theoretical Comparison

SCP GZOCP PhO CAN UCP ≻ ≻ ≻ ≻ ≻ ≻

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SLIDE 71

Theoretical Comparison

SCP GZOCP PhO CAN UCP ≻ ≻ ≻ ≻ ≻ ≻ ≻ ≻ ≻

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SLIDE 72

Theoretical Comparison

SCP GZOCP PhO CAN OUCP UCP ≻ ≻ ≻ ≻ ≻ ≻ ≻ ≻ ≻

31/35

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SLIDE 73

Theoretical Comparison

SCP GZOCP PhO CAN OUCP UCP ≻ ≻ ≻ ≻ ≻ ≻ ≻ ≻ ≻ ≻ ≻ ≻

31/35

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SLIDE 74

Theoretical Comparison

SCP GZOCP PhO CAN OUCP UCP ≻ ≻ ≻ ≻ ≻ ≻ ≻ ≻ ≻ ≻ ≻ ≻

31/35

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SLIDE 75

Experimental Comparison

  • Heuristics: Cartesian abstraction heuristics + PDBs

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SLIDE 76

Experimental Comparison

SCP GZOCP PhO CAN OUCP UCP ≻ ≻ ≻ ≻ ≻ ≻ ≻ ≻ ≻ ≻ ≻ ≻

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SLIDE 77

Experimental Comparison

SCP GZOCP PhO CAN OUCP ≻ ≻ ≻ ≻ ≻ ≻ ≻ ≻ ≻

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SLIDE 78

Experimental Comparison

SCP PhO CAN OUCP ≻ ≻ ≻

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SLIDE 79

Experimental Comparison

SCP PhO CAN ≻ ≻ ≻

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SLIDE 80

Experimental Comparison

SCP

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SLIDE 81

Solved Tasks

700 800 900 1,000 1,100 PhO-Sys2 M&S iPDB

737 808 881

700 800 900 1,000 1,100 CEGAR goals landmarks LMs+goals greedy

  • ptimized

multiple diverse

706 774 785 798 866 881 982 994

34/35

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SLIDE 82

Solved Tasks

700 800 900 1,000 1,100 PhO-Sys2 M&S iPDB

737 808 881

700 800 900 1,000 1,100 CEGAR goals landmarks LMs+goals greedy

  • ptimized

multiple diverse Cart.+PDBs

706 774 785 798 866 881 982 994 1,063

34/35

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SLIDE 83

Conclusion

Counterexample-guided Cartesian Abstraction Refinement

  • refines abstraction only where needed
  • decompositions yield complementary heuristics

Saturated Cost Partitioning

  • assigns each heuristic only the costs it needs
  • best results for diverse optimized orders

Comparison of Cost Partitioning Algorithms

  • dominances and non-dominances
  • saturated cost partitioning preferable in all settings

35/35

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SLIDE 84

Conclusion

Counterexample-guided Cartesian Abstraction Refinement

  • refines abstraction only where needed
  • decompositions yield complementary heuristics

Saturated Cost Partitioning

  • assigns each heuristic only the costs it needs
  • best results for diverse optimized orders

Comparison of Cost Partitioning Algorithms

  • dominances and non-dominances
  • saturated cost partitioning preferable in all settings

35/35

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SLIDE 85

Conclusion

Counterexample-guided Cartesian Abstraction Refinement

  • refines abstraction only where needed
  • decompositions yield complementary heuristics

Saturated Cost Partitioning

  • assigns each heuristic only the costs it needs
  • best results for diverse optimized orders

Comparison of Cost Partitioning Algorithms

  • dominances and non-dominances
  • saturated cost partitioning preferable in all settings

35/35