Faster than Weighted A*: An Optimistic Approach to Bounded Suboptimal Search

Transcription

1 Faster than Weighted A*: An Optimistic Approach to Bounded Suboptimal Search Jordan Thayer and Wheeler Ruml {jtd7, ruml} at cs.unh.edu Jordan Thayer (UNH) Optimistic Search 1 / 45

2 Motivation Motivation Finding optimal solutions is prohibitively expensive. 200,000 Grid Pathfinding A* Grid Pathfinding A* 1.6 Nodes generated 100,000 Solution Cost (relative to A*) , ,000 Problem Size Problem Size Jordan Thayer (UNH) Optimistic Search 2 / 45

3 Motivation Motivation Finding optimal solutions is prohibitively expensive. Greedy solutions can be arbitrarily bad. 200,000 Grid Pathfinding A* Four-way Grid Pathfinding (Unit cost) A* Greedy Greedy 1.6 Nodes generated 100,000 Solution Cost (relative to A*) Problem Size 800 1, Problem Size 800 1,000 Jordan Thayer (UNH) Optimistic Search 3 / 45

4 Motivation Motivation Finding optimal solutions is prohibitively expensive. Greedy solutions can be arbitrarily bad. Weighted A* bounds suboptimality. 200,000 Grid Pathfinding A* wa* Grid Pathfinding A* wa* Greedy Greedy 1.6 Nodes generated 100,000 Solution Cost (relative to A*) , ,000 Problem Size Problem Size Jordan Thayer (UNH) Optimistic Search 4 / 45

5 Motivation Motivation Finding optimal solutions is prohibitively expensive. Greedy solutions can be arbitrarily bad. Weighted A* bounds suboptimality. Optimistic Search: faster search within the same bound. 200,000 Grid Pathfinding A* wa* Optimistic Greedy Grid Pathfinding A* wa* Optimistic Greedy 1.6 Nodes generated 100,000 Solution Cost (relative to A*) Problem Size 800 1, Problem Size 800 1,000 Jordan Thayer (UNH) Optimistic Search 5 / 45

6 Predecessors Basic Idea Jordan Thayer (UNH) Optimistic Search 6 / 45

7 Talk Outline Predecessors Basic Idea Run weighted A with a weight higher than the bound. Expand additional nodes to prove solution quality. The Greedy Search Phase The Cleanup Phase Jordan Thayer (UNH) Optimistic Search 7 / 45

8 Previous Algorithms: A Predecessors Basic Idea A best first search expanding nodes in f order. f(n) = g(n) + h(n) If h(n) is admissible, returns optimal solution. Jordan Thayer (UNH) Optimistic Search 8 / 45

9 Previous Algorithms: Weighted A Predecessors Basic Idea A best first search expanding nodes in f order. f (n) = g(n) + w h(n) Solution quality bounded by w for admissible h(n). Jordan Thayer (UNH) Optimistic Search 9 / 45

10 Optimistic Search: The Basic Idea Predecessors Basic Idea 1. Run weighted A with a high weight. 2. Expand node with lowest f value after a solution is found. Continue until w f min > f(sol) This clean up guarantees solution quality. Jordan Thayer (UNH) Optimistic Search 10 / 45

11 Optimistic Search: The Basic Idea Predecessors Basic Idea 1. Run weighted A with a high weight. 2. Expand node with lowest f value after a solution is found. Continue until w f min > f(sol) This clean up guarantees solution quality. Jordan Thayer (UNH) Optimistic Search 11 / 45

12 Optimistic Search: The Basic Idea Predecessors Basic Idea 1. Run weighted A with a high weight. 2. Expand node with lowest f value after a solution is found. Continue until w f min > f(sol) This clean up guarantees solution quality. Jordan Thayer (UNH) Optimistic Search 12 / 45

13 Optimistic Search: The Basic Idea Predecessors Basic Idea 1. Run weighted A with a high weight. 2. Expand node with lowest f value after a solution is found. Continue until w f min > f(sol) This clean up guarantees solution quality. Jordan Thayer (UNH) Optimistic Search 13 / 45

14 Weighted A Jordan Thayer (UNH) Optimistic Search 14 / 45

15 Talk Outline Weighted A The Greedy Search Phase Weighted A becomes faster as the bound grows. Weighted A is often better than the bound. The Cleanup Phase Jordan Thayer (UNH) Optimistic Search 15 / 45

16 Large Bounds, Faster Solution wa returns solutions faster as the bound increases. Weighted A Pearl and Kim Hard Node Generations Relative to A* wa* Sub-optimality bound 1.2 Jordan Thayer (UNH) Optimistic Search 16 / 45

17 Weighted A is often better than the bound wa returns solutions better than the bound. Weighted A 3 Four-way Grid Pathfinding (Unit cost) y=x wa* Solution Cost (relative to A*) Sub-optimality Bound 3 Jordan Thayer (UNH) Optimistic Search 17 / 45

18 w-admissibility Jordan Thayer (UNH) Optimistic Search 18 / 45

19 Talk Outline w-admissibility The Greedy Search Phase The Cleanup Phase Expand additional nodes in f order. Quit when the solution is provably within the bound. Jordan Thayer (UNH) Optimistic Search 19 / 45

20 Proving w-admissibility w-admissibility p is the deepest node on an optimal path to opt. f min is the node with the smallest f value. Jordan Thayer (UNH) Optimistic Search 20 / 45

21 Proving w-admissibility w-admissibility p is the deepest node on an optimal path to opt. f min is the node with the smallest f value. f(f min ) f(p) f(p) f(opt) f min provides a lower bound on solution cost. Determine f min by priority queue sorted on f Jordan Thayer (UNH) Optimistic Search 20 / 45

22 Proving w-admissibility w-admissibility p is the deepest node on an optimal path to opt. f min is the node with the smallest f value. f(f min ) f(p) f(p) f(opt) f min provides a lower bound on solution cost. Determine f min by priority queue sorted on f Optimistic Search: Run a greedy search Expand f min until w f min f(sol) Jordan Thayer (UNH) Optimistic Search 20 / 45

23 Proving w-admissibility w-admissibility p is the deepest node on an optimal path to opt. f min is the node with the smallest f value. f(f min ) f(p) f(p) f(opt) f min provides a lower bound on solution cost. Determine f min by priority queue sorted on f Optimistic Search: Run a greedy search Expand f min until w f min f(sol) Jordan Thayer (UNH) Optimistic Search 21 / 45

24 Performance Jordan Thayer (UNH) Optimistic Search 22 / 45

25 Talk Outline Performance The Greedy Search Guaranteeing solution quality Results in several domains. Jordan Thayer (UNH) Optimistic Search 23 / 45

26 Performance Sliding Tile Puzzles Korf s puzzle instances (add date) Traveling Salesman Unit Square Pearl and Kim Hard (add date) Grid world path finding Four-way and Eight-way Movement Unit and Life Cost Models 25%, 30%, 35%, 40%, 45% obstacles Temporal Planning Blocksworld, Logistics, Rover, Satellite, Zenotravel See paper for additional plots. Jordan Thayer (UNH) Optimistic Search 24 / 45

27 Performance of Optimistic Search Korf s 15 Puzzles: h = Manhattan Distance Performance Node Generations Relative to IDA* wa* Optimistic Sub-optimality bound Jordan Thayer (UNH) Optimistic Search 25 / 45

28 Performance of Optimistic Search TSP: Pearl and Kim Hard Performance Node Generations Relative to A* wa* Optimistic Sub-optimality bound 1.2 Jordan Thayer (UNH) Optimistic Search 26 / 45

29 Performance of Optimistic Search Performance Nodes generated (relative to A*) Four-way Grid Pathfinding (Unit cost) wa* Optimistic Sub-optimality Bound 3 Jordan Thayer (UNH) Optimistic Search 27 / 45

30 Performance of Optimistic Search logistics (problem 3) Performance Nodes generated (relative to A*) wa* Optimistic Search Sub-optimality Bound 3 Jordan Thayer (UNH) Optimistic Search 28 / 45

31 Expansion Policy BAwA Jordan Thayer (UNH) Optimistic Search 29 / 45

32 Talk Outline Expansion Policy BAwA The Greedy Search Guaranteeing solution quality Strict vs. Loose Expansion Policy Bounded Anytime Weighted A Jordan Thayer (UNH) Optimistic Search 30 / 45

33 Expansion Policy Expansion Policy BAwA Strict Expansion Order: Algorithms like wa, A ǫ, Dynamically Weighted A Any expanded node can be shown to be within the bound at the time of their expansion Quality bound comes from this Loose Expansion Order: Algorithms like Optimistic Search No restriction on the nodes expanded initially. Quality bound requires node expansion beyond the initial solution. Jordan Thayer (UNH) Optimistic Search 31 / 45

34 Bounded Anytime Weighted A Anytime Heuristic Search: Running weighted A with a high weight Continue node expansions after a solution is found Expansion Policy BAwA Jordan Thayer (UNH) Optimistic Search 32 / 45

35 Bounded Anytime Weighted A Expansion Policy BAwA Anytime Heuristic Search: Running weighted A with a high weight Continue node expansions after a solution is found Bounded Anytime Weighted A : Running weighted A with a high weight Continue node expansions after a solution is found Add a second priority queue allows us to converge on a bound instead of on optimal. Jordan Thayer (UNH) Optimistic Search 32 / 45

36 Optimistic Search expansions 1. Run weighted A with a high weight. 2. Expand node with lowest f value after a solution is found. Continue until w f min > f(sol) This clean up guarantees solution quality. Expansion Policy BAwA Jordan Thayer (UNH) Optimistic Search 33 / 45

37 Bounded Anytime Weighted A Expansions 1. Run weighted A with a high weight. 2. Expand node with lowest f value after a solution is found. Continue until w f min > f(sol) This clean up guarantees solution quality. Expansion Policy BAwA Jordan Thayer (UNH) Optimistic Search 34 / 45

38 Bounded Anytime Weighted A* Korf s 15 Puzzles Expansion Policy BAwA Node Generations Relative to IDA* BAwA* wa* Optimistic Sub-optimality bound Jordan Thayer (UNH) Optimistic Search 35 / 45

39 Bounded Anytime Weighted A* Pearl and Kim Hard Expansion Policy BAwA Node Generations Relative to A* BAwA* wa* Optimistic Sub-optimality bound 1.2 Jordan Thayer (UNH) Optimistic Search 36 / 45

40 Advertising Optimistic Search: Simple to implement. Performance is predictable. Current results are good, tuning could help. Optimal greediness is still an open question. Consistently better than Weighted A If you currently use wa, you should use Optimistic Search. Jordan Thayer (UNH) Optimistic Search 37 / 45

41 The University of New Hampshire Advertising Tell your students to apply to grad school in CS at UNH! friendly faculty funding individual attention beautiful campus low cost of living easy access to Boston, White Mountains strong in AI, infoviz, networking, systems, bioinformatics Jordan Thayer (UNH) Optimistic Search 38 / 45

42 Additional Slides Loose Bounds Duplicates Pseudo Code Additional Slides Jordan Thayer (UNH) Optimistic Search 39 / 45

43 Weighted A is often better than the bound wa returns solutions better than the bound. 3 Four-way Grid Pathfinding (Unit cost) y=x wa* Additional Slides Loose Bounds Duplicates Pseudo Code Solution Cost (relative to A*) Sub-optimality Bound 3 Jordan Thayer (UNH) Optimistic Search 40 / 45

44 Weighted A Respects a Bound f(n) = g(n) + h(n) f (n) = g(n) + w h(n) Additional Slides Loose Bounds Duplicates Pseudo Code g(sol) f (sol) f (p) g(p) + w h(p) w (g(p) + h(p)) w f(p) w f(opt) w g(opt) Therefore, g(sol) w g(opt) Jordan Thayer (UNH) Optimistic Search 41 / 45

45 Weighted A Respects the Bound and Then Some f(n) = g(n) + h(n) f (n) = g(n) + w h(n) Additional Slides Loose Bounds Duplicates Pseudo Code g(sol) f (sol) f (p) g(p) + w h(p) w (g(p) + h(p)) w f(p) w f(opt) w g(opt) g(p) + w h(p) w g(p) + w h(p) Jordan Thayer (UNH) Optimistic Search 42 / 45

46 Duplicate Dropping can be Important 0.9 Four-way Grid Pathfinding (Unit cost) wa* wa* dd Additional Slides Loose Bounds Duplicates Pseudo Code Nodes generated (relative to A*) Sub-optimality Bound 3 Jordan Thayer (UNH) Optimistic Search 43 / 45

47 Sometimes it isn t Korf s 15 puzzles Additional Slides Loose Bounds Duplicates Pseudo Code Node Generations Relative to IDA* wa* dd wa* Sub-optimality bound Jordan Thayer (UNH) Optimistic Search 44 / 45

48 Sometimes it isn t Duplicates can be delayed during the greedy search phase. Korf s 15 puzzles Additional Slides Loose Bounds Duplicates Pseudo Code Node Generations Relative to IDA* wa* dd wa* Sub-optimality bound Jordan Thayer (UNH) Optimistic Search 44 / 45

49 Pseudo Code Additional Slides Loose Bounds Duplicates Pseudo Code Optimistic Search(initial, bound) 1. open f {initial} 2. open f {initial} 3. incumbent 4. repeat until bound f(first on open f ) f(incumbent): 5. if f(first on open f) < f(incumbent) then 6. n remove first on open f 7. remove n from open f 8. else n remove first on open f 9. remove n from open f 10. add n to closed 11. if n is a goal then 12. incumbent n 13. else for each child c of n 14. if c is duplicated in open f then 15. if c is better than the duplicate then 16. replace copies in open f and open f 17. else if c is duplicated in closed then 18. if c is better than the duplicate then 19. add c to open f and open f 20. else add c to open f and open f Jordan Thayer (UNH) Optimistic Search 45 / 45