Bounded Suboptimal Search: A Direct Approach Using Inadmissible Estimates

Transcription

1 Bounded Suboptimal Search: A Direct Approach Using Inadmissible Estimates Jordan T. Thayer and Wheeler Ruml jtd7, ruml at cs.unh.edu With Thanks To NSF-IIS and DARPA CSSG N10AP20029 Jordan T. Thayer (UNH) Bounded Suboptimal Search 1 / 19

2 Greedy Search Greedy Search Bounded Search Three Ideas total nodes generated relative to A* Grid Four-way 35% A* greedy final sol cost relative to A* greedy A* Grid Four-way 35% Size Size 800 sacrificing optimality can speed search solutions could be arbitrarily bad Jordan T. Thayer (UNH) Bounded Suboptimal Search 2 / 19

3 Bounded Suboptimal Search: A Middle Ground Greedy Search Bounded Search Three Ideas total nodes generated relative to A* Grid Four-way 35% A* wa* greedy final sol cost relative to A* greedy wa* A* Grid Four-way 35% Size Size 800 given a suboptimality bound w, find a solution with cost within a factor w of optimal as quickly as possible. Jordan T. Thayer (UNH) Bounded Suboptimal Search 3 / 19

4 Three Useful Ideas finding solutions and proving bounds are separate tasks Greedy Search Bounded Search Three Ideas inadmissible cost estimates can be more informed searching on distance is faster than cost Jordan T. Thayer (UNH) Bounded Suboptimal Search 4 / 19

5 Three Useful Ideas finding solutions and proving bounds are separate tasks Greedy Search Bounded Search Three Ideas inadmissible cost estimates can be more informed searching on distance is faster than cost A ǫ Pearl and Kim, 1982 Optimistic Search Thayer and Ruml, 2008 Skeptical Search Thayer and Ruml, 2011 Jordan T. Thayer (UNH) Bounded Suboptimal Search 4 / 19

6 Three Useful Ideas finding solutions and proving bounds are separate tasks Greedy Search Bounded Search Three Ideas inadmissible cost estimates can be more informed searching on distance is faster than cost Heavy Vacuum admiss. h inadmiss. h 400 total raw cpu time Size 30 Jordan T. Thayer (UNH) Bounded Suboptimal Search 4 / 19

7 Three Useful Ideas finding solutions and proving bounds are separate tasks Greedy Search Bounded Search Three Ideas inadmissible cost estimates can be more informed searching on distance is faster than cost Heavy Vacuum admiss. h inadmiss. h d 400 total raw cpu time Size 30 Jordan T. Thayer (UNH) Bounded Suboptimal Search 4 / 19

8 Three Useful Ideas finding solutions and proving bounds are separate tasks Greedy Search Bounded Search Three Ideas inadmissible cost estimates can be more informed searching on distance is faster than cost Explicit Estimation Search () combines these three ideas. Jordan T. Thayer (UNH) Bounded Suboptimal Search 4 / 19

9 Direct Approach Three Heuristics Explicit Estimation Search Jordan T. Thayer (UNH) Bounded Suboptimal Search 5 / 19

10 A Direct Approach minimize solving time subject to suboptimality bound w Direct Approach Three Heuristics Jordan T. Thayer (UNH) Bounded Suboptimal Search 6 / 19

11 A Direct Approach Direct Approach Three Heuristics minimize solving time subject to suboptimality bound w weighted A* (f (n) = g(n)+w h(n)) is simple but ad hoc (Pohl, AIJ vol 1, 1970) Jordan T. Thayer (UNH) Bounded Suboptimal Search 6 / 19

12 A Direct Approach Direct Approach Three Heuristics minimize solving time subject to suboptimality bound w weighted A* (f (n) = g(n)+w h(n)) is simple but ad hoc (Pohl, AIJ vol 1, 1970) expand the node closest to a solution within the bound best d: node estimated within bound closest to a goal Jordan T. Thayer (UNH) Bounded Suboptimal Search 6 / 19

13 Three Heuristic Sources Of Information Direct Approach Three Heuristics 1. h: an admissible estimate of cost-to-go f(n) = g(n)+h(n) finding solutions and proving bounds are separate tasks Jordan T. Thayer (UNH) Bounded Suboptimal Search 7 / 19

14 Three Heuristic Sources Of Information Direct Approach Three Heuristics 1. h: an admissible estimate of cost-to-go f(n) = g(n)+h(n) finding solutions and proving bounds are separate tasks 2. ĥ: a potentially inadmissible estimate of cost-to-go inadmissible cost estimates can be more informed f(n) = g(n)+ĥ(n) (Thayer and Ruml, ICAPS-11) Jordan T. Thayer (UNH) Bounded Suboptimal Search 7 / 19

15 Three Heuristic Sources Of Information Direct Approach Three Heuristics 1. h: an admissible estimate of cost-to-go f(n) = g(n)+h(n) finding solutions and proving bounds are separate tasks 2. ĥ: a potentially inadmissible estimate of cost-to-go inadmissible cost estimates can be more informed f(n) = g(n)+ĥ(n) (Thayer and Ruml, ICAPS-11) 3. d: a potentially inadmissible estimate of distance-to-go searching on distance is faster than cost (Pearl and Kim, IEEE PAMI 1982, Thayer et al, ICAPS-09) Jordan T. Thayer (UNH) Bounded Suboptimal Search 7 / 19

16 Finding best d Direct Approach Three Heuristics best f : open node with minimum f argmin n open f(n) Jordan T. Thayer (UNH) Bounded Suboptimal Search 8 / 19

17 Finding best d Direct Approach Three Heuristics best f : open node with minimum f argmin n open best f: open node with minimum f f(n) argmin n open f(n) Jordan T. Thayer (UNH) Bounded Suboptimal Search 8 / 19

18 Finding best d Direct Approach Three Heuristics best f : open node with minimum f argmin n open best f: open node with minimum f f(n) argmin n open f(n) pursuing the shortest solution within the bound should be fast best d: estimated w-admissible node with minimum d argmin d(n) n open f(n) w f(best f) Jordan T. Thayer (UNH) Bounded Suboptimal Search 8 / 19

19 Expansion Order Direct Approach Three Heuristics best f : open node with minimum f best f: open node with minimum f best d: estimated w-admissible node with minimum d node to expand next: 1. pursue the shortest solution that is within the bound in other words: 1. best d Jordan T. Thayer (UNH) Bounded Suboptimal Search 9 / 19

20 Expansion Order Direct Approach Three Heuristics best f : open node with minimum f best f: open node with minimum f best d: estimated w-admissible node with minimum d node to expand next: 1. pursue the shortest solution that is within the bound in other words: 1. if f(best d) w f(best f ) then best d note that f(best f ) f(opt) and f(n) f(n) Jordan T. Thayer (UNH) Bounded Suboptimal Search 9 / 19

21 Expansion Order Direct Approach Three Heuristics best f : open node with minimum f best f: open node with minimum f best d: estimated w-admissible node with minimum d node to expand next: 1. pursue the shortest solution that is within the bound. 2. pursue the optimal solution. 3. in other words: 1. if f(best d) w f(best f ) then best d 2. else if f(best f) w f(best f ) then best f 3. Jordan T. Thayer (UNH) Bounded Suboptimal Search 9 / 19

22 Expansion Order Direct Approach Three Heuristics best f : open node with minimum f best f: open node with minimum f best d: estimated w-admissible node with minimum d node to expand next: 1. pursue the shortest solution that is within the bound. 2. pursue the optimal solution. 3. raise the lower bound on optimal solution cost. in other words: 1. if f(best d) w f(best f ) then best d 2. else if f(best f) w f(best f ) then best f 3. else best f see paper for further justification Jordan T. Thayer (UNH) Bounded Suboptimal Search 9 / 19

23 Results Dock Robot Direct Approach Three Heuristics 300 total raw cpu time A* eps wa* Optimistic Skeptical Opt. 1.6 Suboptimality 2.4 Jordan T. Thayer (UNH) Bounded Suboptimal Search 10 / 19

24 Results Direct Approach Three Heuristics log10 total raw cpu time Vacuum World A* eps Optimistic Skeptical wa* Opt Suboptimality Jordan T. Thayer (UNH) Bounded Suboptimal Search 10 / 19

25 Results Heavy Vacuum World Direct Approach Three Heuristics log10 total raw cpu time 1 0 wa* Optimistic Skeptical A* eps Opt Suboptimality Jordan T. Thayer (UNH) Bounded Suboptimal Search 10 / 19

26 Contributions Explicit Estimation Search () follows directly from the objectives of bounded suboptimal search state of the art search bounded suboptimal search use inadmissible heuristics without losing bounds robust, works best in domains with action costs Jordan T. Thayer (UNH) Bounded Suboptimal Search 11 / 19

27 The University of New Hampshire 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 T. Thayer (UNH) Bounded Suboptimal Search 12 / 19

28 Nodes Backup Slides Nodes Bound Overhead A ǫ A ǫ Failure best f = argmin n open best d = best f = argmin n open f(n) argmin n open f(n) w f(best f) f(n) d(n) Jordan T. Thayer (UNH) Bounded Suboptimal Search 13 / 19

29 Respects a Bound Backup Slides Nodes Bound Overhead A ǫ A ǫ Failure p is the deepest node on an optimal path to opt. best f isthenodewiththe smallest f value. f(p) f(opt) f(best f ) f(p) best f provides a lower bound on solution cost. determine best f by priority queue sorted on f Jordan T. Thayer (UNH) Bounded Suboptimal Search 14 / 19

30 Why Doesn t A ǫ Work Well? Backup Slides Nodes Bound Overhead A ǫ A ǫ Failure Jordan T. Thayer (UNH) Bounded Suboptimal Search 15 / 19

31 Overhead 3 Life Four-way Grid World Skeptical wa* 8 Life Four-way Grid World Skeptical wa* Backup Slides Nodes Bound Overhead A ǫ A ǫ Failure log10 total raw cpu time 2 1 log10 total nodes generated Suboptimality 4 2 Suboptimality 4 Jordan T. Thayer (UNH) Bounded Suboptimal Search 16 / 19

32 A ǫ Pearl and Kim, 1982 intuition: of all solutions within the bound, the nearest should be the fastest to find. Backup Slides Nodes Bound Overhead A ǫ A ǫ Failure f(n) = g(n)+h(n) best f : generated but unexpanded node with minimum f best-first search on two lists: open: all generated but unexpanded nodes, sorted on f(n) focal: all nodes where f(n) w f(best f ) sorted on d(n) expand the best node from focal Jordan T. Thayer (UNH) Bounded Suboptimal Search 17 / 19

33 A ǫ Doesn t Work Very Well Thayer et al SoCS-09 Life Four-way Grid World A* eps Backup Slides Nodes Bound Overhead A ǫ A ǫ Failure total raw cpu time relative to A* Suboptimality 4 Jordan T. Thayer (UNH) Bounded Suboptimal Search 18 / 19

34 Why Doesn t A ǫ Work Well? Thayer et al SoCS-09 open: all generated but unexpanded nodes, sorted on f(n) focal: all nodes where f(n) w f(best f ) sorted on d(n) Backup Slides Nodes Bound Overhead A ǫ A ǫ Failure open best f f often d focal d best d often f Jordan T. Thayer (UNH) Bounded Suboptimal Search 19 / 19

35 Why Doesn t A ǫ Work Well? Thayer et al SoCS-09 open: all generated but unexpanded nodes, sorted on f(n) focal: all nodes where f(n) w f(best f ) sorted on d(n) Backup Slides Nodes Bound Overhead A ǫ A ǫ Failure open best f f often d best d focal d best d best f often f Jordan T. Thayer (UNH) Bounded Suboptimal Search 19 / 19

36 Why Doesn t A ǫ Work Well? Thayer et al SoCS-09 open: all generated but unexpanded nodes, sorted on f(n) focal: all nodes where f(n) w f(best f ) sorted on d(n) Backup Slides Nodes Bound Overhead A ǫ A ǫ Failure open best f f often d best d focal d best d best f often f f rises as search progresses (h is admissible) best d s children won t remain on focal Jordan T. Thayer (UNH) Bounded Suboptimal Search 19 / 19