COMP 8620 Advanced Topics in AI

Transcription

1 OMP 8620 dvanced Topics in Lecturers: Philip Kilby & Jinbo uang 25/07/2008 1

2 Part 1: Search Lecturer: r Philip Kilby Philip.Kilby@nicta.com.au Weeks 1-71 (Week 7 is assignment seminars) Part 2: Probabilistic Reasoning with ayesian networks Lecturer: r Jinbo uang Jinbo.uang@nicta.com.au Weeks /07/2008 2

3 ourse Outline Part ntroduction Systematic Search 3-4 Linear Programming / nteger Programming / ranch and ound 5-6 Neighbourhood-based based methods: Simulated nnealing Tabu Search enetic lgorithms 7-8 onstraint Programming (uest lecturer: Jason Li) 7-8 onstraint Programming (uest lecturer: Jason Li) 9-10 ase Studies: TSP Planning (Jussi( Rintanen) 11 ynamic Programming 12 Qualitative Reasoning (uest lecturer: Jason Li) Seminars (uest Lecturers: You lot!) 3

4 40% xam 60% ssignment ssessment Search 1 x ssignment (next week) 15% 1 x Seminar 15% Reasoning 3 x ssignment (10% each) (tentative) 4

5 What is Search? Search finds a solution to a problem Landscape of solutions Some are good Lots (and lots and lots) are bad ow do we find the good ones? 5

6 What is Search? Landscape can continuous or discrete (We will only deal with discrete) Landscape can be (approximated by?) a tree Landscape can be (approximated by?) a graph 6

7 What is Search? X + 2 * Y < 3, X [0,5] Y [1,5] 7

8 What is Search? X=2 Y=5 X=3 Y=5 X + 2 * Y < 3, X [0,5] Y [1,5] X=2 Y=3 X=0 Y=0 X=5 Y=5 X=5 Y=1 X=0 Y=1 X=3 Y=1 X=4 Y=5 X=1 Y=2 X=0 Y=2 X=0 Y=3 X=5 Y=2 X=0 Y=1 X=3 Y=2 X=2 Y=2 X=5 Y=3 X=3 Y=3 X=4 Y=3 X=1 Y=1 X=3 Y=4 X=2 Y=1 X=0 Y=4 X=5 Y=4 X=0 Y=5 X=4 Y=4 X=4 Y=2 X=1 Y=5 X=1 Y=4 X=1 Y=3 X=2 Y=4 8 X=4 Y=1

9 What is Search? X + 2 * Y < 3, X [0,5] Y [1,5] X=0 X=1 X=2 X=3 X=4 X=5 Y=1 Y=2 Y=3 Y=4 Y=5 9

10 What is Search? X + 2 * Y < 3, X [0,5] Y [1,5] Y=1 Y=2 Y=3 Y=4 Y=5 X=0 X=1 X=2 X=3 X=4 X=5 10

11 What is Search? X + 2 * Y < 3, X [0,5] Y [1,5] X>0 X=0 Y>1 Y=1 11

12 What is Search? ifferent flavours ind a solution (Satisfaction/ecision) ind the best solution (ombinatorial Optimisation) ecision problems: Search variant: ind a solution (or show none exists) ecision variant: oes a solution exist? 12

13 What is Search onstructive Search ind a solution by construction Local Search iven a solution, explore the neighbourhood of that solution to find other (possibly better) solutions 13

14 What is Search? nytime algorithm (for optimisation prob): fter a (short) startup time, an answer is available The longer the alg is run, the better the solution Quality guarantee may be available (e.g. solution is within 5% of optimal) One-shot algorithm nswer only available at completion 14

15 What is Search? problem is defined by nitial state Successor function (an action leading from one state to another) oal test Path cost solution is a sequence of actions leading from the start state to a goal state 15

16 xample: Romania 16

17 Problem formulation problem is defined by four items: initial state e.g., at rad successor function S(x) set of action state pairs e.g., S(rad) = {<rad Zerind, at Zerind>,...} goal test e.g., x = at ucharest can be implicit, e.g., asirport(x) path cost (additive) e.g. sum of distances, number of actions executed, etc. c(x, a, y) is the step cost, assumed to be 0 17

18 Tree Search function Tree-Search (problem, fringe) returns a solution, or failure fringe nsert (Make-Node (nitial-state (problem)), fringe) loop do if s-mpty (fringe) then return failure node Remove-ront (fringe) if oal-test (problem, State[node]) then return node fringe nsertll (xpand (node, problem), fringe) function xpand (node, problem) returns a set of nodes successors the empty set for each (action, result) in Successor-n (problem, State[node]) do s a new Node Parent-Node[s] node; ction[s] action; State[s] result Path-ost[s] Path-ost[node] + Step-ost(State[node], action, result) epth[s] epth[node] + 1 successors nsert (s) return successors 18

19 readth-first Search L M O 19

20 readth-first Search L M O 20

21 readth-first Search L M O 21

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33 epth-first Search L M O 33

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36 epth-first Search L M O 36

37 epth-first Search L M O 37

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40 epth-first Search L M O 40

41 epth-first Search L M O 41

42 epth-first Search L M O 42

43 epth-first Search L M O 43

44 epth-first Search L M O 44

45 epth-first Search L M O 45

46 readth-first Search L M!! O 46

47 terative eepening S Search L M O 47

48 terative eepening S Search L M O 48

49 terative eepening S Search L M O 49

50 terative eepening S Search L M O 50

51 terative eepening S Search L M O 51

52 terative eepening S Search L M O 52

53 terative eepening S Search L M O 53

54 terative eepening S Search L M O 54

55 terative eepening S Search L M O 55

56 terative eepening S Search L M O 56

57 terative eepening S Search L M O 57

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61 terative eepening S Search L M O 61

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66 terative eepening S Search L M O 66

67 terative eepening S Search L M O 67

68 terative eepening S Search L M O 68

69 terative eepening S Search L M O 69

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71 terative eepening S Search L M O 71

72 terative eepening S Search L M O 72

73 terative eepening S Search L M O 73

74 terative eepening S Search L M!! O 74

75 Uniform cost search xpand in order of cost dentical to readth-irst Search if all step costs are equal Number of nodes expanded depends on ε > 0 the smallest cost action cost. (f ε = 0, uniform-ost search may infinite-loop) 75

76 i-directional Search xtend S from Start and oal states simultaneously b d/2 d/2 + b d/2 << b d 76

77 i-directional Search Restrictions: an only use if 1 (or a few) known goal nodes an only use if actions are reversible i.e. if we can efficiently calculate Pred(s (s) (for state s) where a Pred( s) = { r : a r s} 77

78 Summary riterion omplete S Yes S No S Yes Uniform ost Yes i-direc direc- tional Yes Time O(b d+1 ) O(b m ) O(b d ) O(b */ */ε ) O(b d/2 ) Space O(b d+1 ) O(bm) O(bd) O(b */ */ε ) O(b d/2 ) Optimal Yes, for unit cost No Yes, for unit cost Yes Yes, for unit cost b = branching factor ; m = max depth of tree ; d = depth of goal node * = cost of goal ; ε = min action cost 78

79 Repeated State etection Test for previously-seen seen states ssentially, search graph mimics state graph 79

80 Repeated State etection an make exponential improvement in time an make S complete S now has potentially exponential space requirement ave to be careful to guarantee optimality n practice, almost always use repeated state detection 80

81 * Search dea: avoid expanding paths that are already expensive valuation function f(n) = g(n) + h(n) g(n) = cost so far to reach n h(n) = estimated cost from n to the closest goal f(n) = estimated total cost of path through n to goal search uses an admissible heuristic i.e., h(n) h (n) where h (n) is the true cost from n. (lso require h(n) 0, so h() = 0 for any goal.).g., h SL (n) never overestimates the actual road distance When h(n) is admissible, f(n) never overestimates the total cost of the shortest path through n to the goal Theorem: search is optimal 81

82 * xample rad 366=

83 * xample rad 366=

84 * xample rad Sibiu 393= Timisoara 447= Zerind 449=

85 * xample rad Sibiu 393= Timisoara 447= Zerind 449=

86 * xample rad Sibiu Timisoara 447= Zerind 449= rad 646= agaras 415= Oradea 671= Rimnicu Vilcea 413=

87 * xample rad Sibiu Timisoara 447= Zerind 449= rad 646= agaras 415= Oradea 671= Rimnicu Vilcea 413=

88 * xample rad Sibiu Timisoara 447= Zerind 449= rad 646= agaras 415= Oradea 671= Rimnicu Vilcea raiova 526= Pitesti 417= Sibiu 553=

89 * xample rad Sibiu Timisoara 447= Zerind 449= rad 646= agaras 415= Oradea 671= Rimnicu Vilcea raiova 526= Pitesti 417= Sibiu 553=

90 * xample rad Sibiu Timisoara 447= Zerind 449= rad 646= agaras Oradea 671= Rimnicu Vilcea Sibiu 591= ucharest 450=450+0 raiova 526= Pitesti 417= Sibiu 553=

91 * xample rad Sibiu Timisoara 447= Zerind 449= rad 646= agaras Oradea 671= Rimnicu Vilcea Sibiu 591= ucharest 450=450+0 raiova 526= Pitesti 417= Sibiu 553=

92 * xample rad Sibiu Timisoara 447= Zerind 449= rad 646= agaras Oradea 671= Rimnicu Vilcea Sibiu 591= ucharest 450=450+0 raiova 526= Pitesti Sibiu 553= ucharest 418=418+0 raiova 615= Rimnicu Vilcea 607=

93 * xample rad Sibiu Timisoara 447= Zerind 449= rad 646= agaras Oradea 671= Rimnicu Vilcea Sibiu 591= ucharest 450=450+0 raiova 526= Pitesti Sibiu 553= ucharest 418=418+0 raiova 615= Rimnicu Vilcea 607=

94 * xample rad Sibiu Timisoara 447= Zerind 449= rad 646= agaras Oradea 671= Rimnicu Vilcea Sibiu 591= ucharest 450=450+0 raiova 526= Pitesti Sibiu 553= ucharest 418=418+0 raiova 615= Rimnicu Vilcea 607=

95 * Search Optimal No other optimal algorithm expands fewer nodes (modulo order of expansion of nodes with same cost) xponential space requirement 95

96 reedy Search Use (always admissible) h(n) = 0 in f(n) = g(n) + h(n) i.e. always use the cheapest found so far 96

97 Variants of * Search Many variants of * developed Weighted *: f(n) = g(n) + W h(n) Not exact aster nswer is within W of optimal 97

98 Variants of * Search Many variants of * developed Weighted *: f(n) = g(n) + W h(n) Not exact aster nswer is within W of optimal * 98

99 Variants of * Search Many variants of * developed Weighted *: f(n) = g(n) + W h(n) Not exact aster nswer is within W of optimal Weighted * 99

100 Variants of * Search terative eepening * (*) ncrementally increase max depth Memory-bounded * mpose bound on memory iscard nodes with largest f(x) if necessary Others to come in seminars 100

101 Limited iscrepancy Search Requires a heuristic to order choices (e.g. admissible heuristic, but don t t require underestimate) f heuristic was right, we could always take the choice given by the heuristic LS allows the heuristic to make mistakes: k-ls allows k mistakes. 101

102 Limited iscrepancy Search (k( = 1) J K L M N O 102

103 Limited iscrepancy Search (k( = 1) 0 1 J K L M N O 103

104 Limited iscrepancy Search (k( = 1) J K L M N O 104

105 Limited iscrepancy Search (k( = 1) J K L M N O 105

106 Limited iscrepancy Search (k( = 1) J K L M N O 106

107 Limited iscrepancy Search (k( = 1) J K L M N O 107

108 Limited iscrepancy Search (k( = 1) J K L M N O 108

109 Limited iscrepancy Search (k( = 1) J K L M N O 109

110 Limited iscrepancy Search (k( = 1) J K L M N O 110

111 Limited iscrepancy Search (k( = 1) J K L M N O 111

112 Limited iscrepancy Search (k( = 1) J K L M N O 112

113 Next week. (nteger) Linear Programming and ranch & ound 113