Lookahead pathology in real-time pathfinding

1. Lookahead pathology in real-time pathfinding Mitja Luštrek Jožef Stefan Institute, Department of Intelligent Systems Vadim Bulitko University of Alberta, Department of Computer Science

2. Introduction • Problem • Explanation

3. Agent-centered search (LRTS) Lookahead area Current state Goal state Lookahead depth d

4. Agent-centered search (LRTS) f = g + h True shortest distance g Estimated shortest distance h Frontier state

5. Agent-centered search (LRTS) Frontier state with the lowest f (fopt)

6. Agent-centered search (LRTS)

7. Agent-centered search (LRTS) h = fopt

8. Agent-centered search (LRTS)

9. Lookahead pathology • Generally believed that larger lookahead depths produce better solutions • Solution-length pathology: larger lookahead depths produce worse solutions Degree of pathology = 2

10. Lookahead pathology • Pathology on states that do not form a path • Error pathology: larger lookahead depths produce more suboptimal decisions Degree of pathology = 2 There is pathology

11. Introduction • Problem • Explanation

12. Our setting • HOG – Hierarchical Open Graph [Sturtevant et al.] • Maps from commercial computer games (Baldur’s Gate, Warcraft III) • Initial heuristic: octile distance (true distance assuming an empty map) • 1,000 problems (map, start state, goal state)

13. On-policy experiments • The agent follows a path from the start state to the goal state, updating the heuristic along the way • Solution length and error over the whole path computed for each lookahead depth -> pathology d = 1 d = 2 d = 3

14. Off-policy experiments • The agent spawns in a number of states • It takes one move towards the goal state • Heuristic not updated • Error is computed from these first moves -> pathology d = 3 d = 1, 2 d = 1 d = 1 d = 2 d = 2, 3 d = 3

15. Basic on-policy experiment • A lot of pathology – over 60%! • First explanation: a lot of states are intrinsically pathological (off-policy mode) • Not true: only 3.9% are • If the topology of the maps is not at fault, perhaps the algorithm is to blame?

16. Off-policy experiment on 188 states • Comparison not fair: • On-policy: pathology from error over a number of states • Off-policy: pathologicalness of single states • Fair: off-policy error over the same number of states as on-policy – 188 (chosen randomly) • Can use only error – no solution length off-policy • Not much less pathology than on-policy: 42.2% vs. 61.5%

17. Tolerance • The first off-policy experiment showed little pathology, the second one quite a lot • Perhaps off-policy pathology is caused by minor differences in error – noise • Introduce tolerence t: • increase in error counts towards the pathology only if error (d1) > t ∙ error (d2) • set t so that the pathology in the off-policy experiment on 188 states is < 5%: t = 1.09

18. Experiments with t = 1.09 • On-policy changes little vs. t = 1: 57.7% vs. 61.9% • Apparently on-policy pathology is more severe than off-policy • Investigate why! • The above experiments are the basic on-policy experiment and the basic off-policy experiment

19. Introduction • Problem • Explanation

20. Hypothesis 1 • LRTS tends to visit pathological states with an above-average frequency • Test: compute pathology from states visited on-policy instead of 188 random states • More pathology than in random states: 6.3% vs. 4.3% • Much less pathology than basic on-policy: 6.3% vs. 57.7% • Hypothesis 1 is correct, but it is not the main reason for on-policy pathology

21. Is learning the culprit? • There is learning (updating the heuristic) on-policy, but not off-policy • Learning necessary on-policy, otherwise the agent gets caught in infinite loops • Test: traverse paths in the normal on-policy manner, measure error without learning • Less pathology than basic on-policy: 20.2% vs. 57.7% • Still more pathology than basic off-policy: 20.2% vs. 4.3% • Learning is a reason, although not the only one

22. Hypothesis 2 • Larger fraction of updated states at smaller depths Current lookahead area Updated state

23. Hypothesis 2 • Smaller lookahead depths benefit more from learning • This makes their decisions better than the mere depth suggests • Thus they are closer to larger depths • If they are closer to larger depths, cases where a larger depth happens to be worse than a smaller depth are more common • Test: equalize depths by learning as much as possible in the whole lookahead area – uniform learning

24. Uniform learning

25. Uniform learning Search

26. Uniform learning Update

27. Uniform learning Search

28. Uniform learning Update

29. Uniform learning

30. Uniform learning

31. Uniform learning

32. Uniform learning

33. Pathology with uniform learning • Even more pathology than basic on-policy: 59.1% vs. 57.7% • Is Hypothesis 2 wrong? • Let us look at the volume of heuristic updates encountered per state generated during search • This seems to be the best measure of the benefit of learning

34. Volume of updates encountered • Hypothesis 2 is correct after all

35. Hypothesis 3 • On-policy: one search every d moves, so fewer searchs at larger depths • Off-policy: one search every move

36. Hypothesis 3 • The difference between depths in the amount of search is smaller on-policy than off-policy • This makes the depths closer on-policy • If they are closer, cases where a larger depth happens to be worse than a smaller depth are more common • Test: search every move on-policy

37. Pathology when searching every move • Less pathology than basic on-policy: 13.1% vs. 57.7% • Still more pathology than basic off-policy: 13.1% vs. 4.3% • Hypothesis 3 is correct, the remaining pathology due to Hypotheses 1 and 2 • Further test: number of states generated per move

38. States generated / move • Hypothesis 3 confirmed again

39. Summary of explanation • On-policy pathology caused by different lookahead depths being closer to each other in terms of the quality of decisions than the mere depths would suggest: • due to the volume of heuristic updates ecnountered per state generated • due to the number of states generated per move • LRTS tends to visit pathological states with an above-average frequency

40. Thank you. Questions?