Informed Search A* Algorithm

Transcription

1 Informed Search A* Algorithm CE417: Introduction to Artificial Intelligence Sharif University of Technology Spring 2018 Soleymani Artificial Intelligence: A Modern Approach, Chapter 3 Most slides have been adopted from Klein and Abdeel, CS188, UC Berkeley.

2 Outline Heuristics Greedy (best-first) search A * search Finding heuristics 2

3 Uninformed Search

4 Uniform Cost Search Strategy: expand lowest path cost c 1 c 2 c 3 The good: UCS is complete and optimal! The bad: Explores options in every direction No information about goal location Start Goal

5 UCS Example 5

6 Informed Search

7 Search Heuristics A heuristic is: A function that estimates how close a state is to a goal Designed for a particular search problem Examples: Manhattan distance, Euclidean distance for pathing

8 Heuristic Function Incorporating problem-specific knowledge in search Information more than problem definition In order to come to an optimal solution as rapidly as possible h n : estimated cost of cheapest path from n to a goal Depends only on n (not path from root to n) If n is a goal state then h(n)=0 h(n) 0 Examples of heuristic functions include using a rule-of-thumb, an educated guess, or an intuitive judgment 8

9 Example: Heuristic Function h(x)

10 Greedy Search

11 Greedy search Priority queue based on h(n) e.g., h SLD n = straight-line distance from n to Bucharest Greedy search expands the node that appears to be closest to goal Greedy 11

12 Example: Heuristic Function h(x)

13 Romania with step costs in km 13

14 Greedy best-first search example 14

15 Greedy best-first search example 15

16 Greedy best-first search example 16

17 Greedy best-first search example 17

18 Greedy Search Expand the node that seems closest What can go wrong?

19 Greedy Search Strategy: expand a node that you think is closest to a goal state Heuristic: estimate of distance to nearest goal for each state b A common case: Best-first takes you straight to the (wrong) goal Worst-case: like a badly-guided DFS b

20 Properties of greedy best-first search Complete? No Time Space Similar to DFS, only graph search version is complete in finite spaces Infinite loops, e.g., (Iasi to Fagaras) Iasi Neamt Iasi Neamt O(b m ), but a good heuristic can give dramatic improvement O(b m ): keeps all nodes in memory Optimal? No 20

21 Greedy Search 21

22 A* Search

23 A * search Idea: minimizing the total estimated solution cost Evaluation function for priority f n = g n + h(n) g n = cost so far to reach n h n = estimated cost of the cheapest path from n to goal So, f n = estimated total cost of path through n to goal Actual cost g n Estimated cost h n start n goal f n = g n + h(n) 24

24 Combining UCS and Greedy Uniform-cost orders by path cost, or backward cost g(n) Greedy orders by goal proximity, or forward cost h(n) 8 1 e h=1 g = 1 h=5 a S g = 0 h=6 1 3 S a d h=6 1 h=5 h=2 1 c b h=7 h=6 A* Search orders by the sum: f(n) = g(n) + h(n) 2 G h=0 g = 2 h=6 g = 3 h=7 b c d G g = 4 h=2 g = 6 h=0 e d G g = 9 h=1 g = 10 h=2 g = 12 h=0 Example: Teg Grenager

25 When should A* terminate? Should we stop when we enqueue a goal? 2 h = 2 A 2 S h = 3 h = 0 G 2 B 3 h = 1 No: only stop when we dequeue a goal

26 Is A* Optimal? h = 6 1 A 3 S h = 7 G h = 0 5 What went wrong? Actual bad goal cost < estimated good goal cost We need estimates to be less than actual costs!

27 Admissible Heuristics

28 Idea: Admissibility Inadmissible (pessimistic) heuristics break optimality by trapping good plans on the frontier Admissible (optimistic) heuristics slow down bad plans but never outweigh true costs

29 Admissible Heuristics A heuristic h is admissible (optimistic) if: where is the true cost to a nearest goal Examples: 15 Coming up with admissible heuristics is most of what s involved in using A* in practice.

30 Optimality of A* Tree Search

31 Optimality of A* Tree Search Assume: A is an optimal goal node B is a suboptimal goal node h is admissible Claim: A will exit the frontier before B

32 Optimality of A* Tree Search: Blocking Proof: Imagine B is on the frontier Some ancestor n of A is on the frontier, too (maybe A!) Claim: n will be expanded before B 1. f(n) is less or equal to f(a) f n g n + h (n) Definition of f-cost Admissibility of h g A = g n + h (n) h = 0 at a goal

33 Optimality of A* Tree Search: Blocking Proof: Imagine B is on the frontier Some ancestor n of A is on the frontier, too (maybe A!) Claim: n will be expanded before B 1. f(n) is less or equal to f(a) 2. f(a) is less than f(b) B is suboptimal h = 0 at a goal

34 Optimality of A* Tree Search: Blocking Proof: Imagine B is on the frontier Some ancestor n of A is on the frontier, too (maybe A!) Claim: n will be expanded before B 1. f(n) is less or equal to f(a) 2. f(a) is less than f(b) 3. n expands before B All ancestors of A expand before B A expands before B A* search is optimal

35 A * search Combines advantages of uniform-cost and greedy searches A * can be complete and optimal when h(n) has some properties 36

36 A * search: example 37

37 A * search: example 38

38 A * search: example 39

39 A * search: example 40

40 A * search: example 41

41 A * search: example 42

42 Properties of A* Uniform-Cost A* b b

43 A* Example 44

44 Example UCS Greedy A* 45

45 UCS vs A* Contours Uniform-cost (A* using h(n)=0) expands equally in all directions Start Goal A* expands mainly toward the goal, but does hedge its bets to ensure optimality More accurate heuristics stretched toward the goal (more narrowly focused around the optimal path) Start Goal States are points in 2-D Euclidean space. g(n)=distance from start h(n)=estimate of distance from goal

46 Comparison Uniform Cost Greedy A*

47 Graph Search

48 Tree Search: Extra Work! Failure to detect repeated states can cause exponentially more work. State Graph Search Tree

49 Graph Search In BFS, for example, we shouldn t bother expanding the circled nodes (why?) S d e p b c e h r q a a h r p q f p q f q c G q c G a a

50 Recall: Graph Search Idea: never expand a state twice How to implement: Tree search + set of expanded states ( closed set ) Expand the search tree node-by-node, but Before expanding a node, check to make sure its state has never been expanded before If not new, skip it, if new add to closed set Important: store the closed set as a set, not a list Can graph search wreck completeness? Why/why not? How about optimality?

51 Optimality of A* Graph Search

52 A* Graph Search Gone Wrong? State space graph Search tree S h=2 1 1 B A h=4 2 1 C h=1 3 S (0+2) A (1+4) B (1+1) C (2+1) C (3+1) h=1 G G (5+0) G (6+0) h=0

53 Conditions for optimality of A * Admissibility: h(n) be a lower bound on the cost to reach goal Condition for optimality of TREE-SEARCH version of A * Consistency (monotonicity): h n c n, a, n + h n Condition for optimality of GRAPH-SEARCH version of A * 54

54 Consistent heuristics Triangle inequality n c(n, a, n ) n h(n) h(n ) G for every node n and every successor n generated by any action a h n c n, a, n + h n c n, a, n : cost of generating n by applying action to n 55

55 Consistency of Heuristics Main idea: estimated heuristic costs actual costs Admissibility: heuristic cost actual cost to goal A h=4 1 C h=1 h=2 3 h(a) actual cost from A to G Consistency: heuristic arc cost actual cost for each arc h(a) h(c) cost(a to C) Consequences of consistency: The f value along a path never decreases h(a) cost(a to C) + h(c) G A* graph search is optimal

56 Optimality Tree search: A* is optimal if heuristic is admissible UCS is a special case (h = 0) Graph search: A* optimal if heuristic is consistent UCS optimal (h = 0 is consistent) Consistency implies admissibility In general, most natural admissible heuristics tend to be consistent, especially if from relaxed problems

57 Consistency implies admissibility Consistency Admissblity All consistent heuristic functions are admissible Nonetheless, most admissible heuristics are also consistent c(n 1, a 1, n 2 ) c(n 2, a 2, n 3 ) c(n k, a k, G) n 1 n 2 n 3 n k G h n 1 c n 1, a 1, n 2 + h(n 2 ) c n 1, a 1, n 2 + c n 2, a 2, n 3 + h(n 3 ) k i=1 c n i, a i, n i+1 + h(g) 0 h n 1 cost of (every) path from n 1 to goal cost of optimal path from n 1 to goal 58

58 Admissible but not consistent: Example G n n g n = 5 h n = 9 f(n) = 14 g n = 6 h n = 6 f(n ) = 12 c(n, a, n ) = 1 h(n) = 9 h(n ) = 6 h n h n + c(n, a, n ) f (for admissible heuristic) may decrease along a path Is there any way to make h consistent? h n = max(h n, h n c(n, a, n )) 59

59 Optimality of A* Graph Search Sketch: consider what A* does with a consistent heuristic: Fact 1: In graph search, A* expands nodes in increasing total f value (f-contours) Fact 2: For every state s, nodes that reach s optimally are expanded before nodes that reach s suboptimally Result: A* graph search is optimal f 1 f 2 f 3

60 Optimality of A * (consistent heuristics) Theorem: If h(n) is consistent, A* using GRAPH-SEARCH is optimal Lemma1: if h(n) is consistent then f(n) values are nondecreasing along any path Proof: Let n be a successor of n I. f(n ) = g(n ) + h(n ) II. g(n ) = g(n) + c(n, a, n ) III. I, II f n = g n + c n, a, n + h n IV. h n is consistent h n c n, a, n + h n V. III, IV f(n ) g(n) + h(n) = f(n) 61

61 Optimality of A * (consistent heuristics) Lemma 2: If A* selects a node n for expansion, the optimal path to that node has been found. Proof by contradiction: Another frontier node n must exist on the optimal path from initial node to n (using graph separation property). Moreover, based on Lemma 1, f n f n and thus n would have been selected first. Lemma 1 & 2 The sequence of nodes expanded by A* (using GRAPH- SEARCH) is in non-decreasing order of f(n) Since h = 0 for goal nodes, the first selected goal node for expansion is an optimal solution (f is the true cost for goal nodes) 62

62 Admissible vs. consistent (tree vs. graph search) Consistent heuristic: When selecting a node for expansion, the path with the lowest cost to that node has been found When an admissible heuristic is not consistent, a node will need repeated expansion, every time a new best (so-far) cost is achieved for it. 63

63 Contours in the state space (using GRAPH-SEARCH) expands nodes in order of increasing f value A * Gradually adds "f-contours" of nodes Contour i has all nodes with f = f i wheref i < f i+1 A* expands all nodes with f(n) < C* A* expands some nodes with f(n) = C* (nodes on the goal contour) A* expands no nodes with f(n) > C* pruning 64

64 Properties of A* Complete? Yes if nodes with f f G = C are finite Time? Step cost ε > 0 and b is finite Exponential But, with a smaller branching factor b h h or when equal step costs b d h h h Polynomial when h(x) h (x) = O(log h (x)) However,A* is optimally efficient for any given consistent heuristic Space? No optimal algorithm of this type is guaranteed to expand fewer nodes than A* (except to node with f = C ) Keeps all leaf and/or explored nodes in memory Optimal? Yes (expanding node in non-decreasing order of f) 65

65 Robot navigation example Initial state? Red cell States? Cells on rectangular grid (except to obstacle) Actions? Move to one of 8 neighbors (if it is not obstacle) Goal test? Green cell Path cost? Action cost is the Euclidean length of movement 66

66 A* vs. UCS: Robot navigation example Heuristic: Euclidean distance to goal Expanded nodes: filled circles in red & green Color indicating g value (red: lower, green: higher) Frontier: empty nodes with blue boundary Nodes falling inside the obstacle are discarded 67 Adopted from:

67 Robot navigation: Admissible heuristic Is Manhattan d M x, y = x 1 y 1 + x 2 y 2 distance an admissible heuristic for previous example? 68

68 A*: inadmissible heuristic h = 5 h_sld h = h_sld Adopted from: 69

69 A*: Summary

70 A*: Summary A* uses both backward costs and (estimates of) forward costs A* is optimal with admissible / consistent heuristics Heuristic design is key: often use relaxed problems

71 Creating Heuristics

72 Creating Admissible Heuristics Most of the work in solving hard search problems optimally is in coming up with admissible heuristics Often, admissible heuristics are solutions to relaxed problems, where new actions are available Inadmissible heuristics are often useful too

73 Example: 8 Puzzle Start State Actions Goal State What are the states? How many states? What are the actions? How many successors from the start state? What should the costs be?

74 8 Puzzle: Heuristic h1 Heuristic: Number of tiles misplaced Why is it admissible? h1(start) = 8 Start State Goal State Average nodes expanded when the optimal path has 4 steps 8 steps 12 steps UCS 112 6, x 10 6 A*(h1) Statistics from Andrew Moore

75 Admissible heuristics: 8-puzzle h 1 (n) = number of misplaced tiles h 2 (n) = sum of Manhattan distance of tiles from their target position i.e., no. of squares from desired location of each tile h 1 (S) = h 2 (S) = = 18 76

76 8 Puzzle: Heuristic h2 What if we had an easier 8-puzzle where any tile could slide any direction at any time, ignoring other tiles? Total Manhattan distance Start State Goal State Why is it admissible? h2(start) = = 18 Average nodes expanded when the optimal path has 4 steps 8 steps 12 steps A*(h1) A*(h2)

77 8 Puzzle: heuristic How about using the actual cost as a heuristic? Would it be admissible? Would we save on nodes expanded? What s wrong with it? With A*: a trade-off between quality of estimate and work per node As heuristics get closer to the true cost, you will expand fewer nodes but usually do more work per node to compute the heuristic itself

78 Effect of heuristic on accuracy N: number of generated nodes by A* d: solution depth Effective branching factor b : branching factor of a uniform tree of depth d containing N + 1 nodes. N + 1 = 1 + b + (b ) (b ) d Well-defined heuristic: b is close to one 79

79 Comparison on 8-puzzle Search Cost (N) d IDS A*(h 1 ) A*(h 2 ) Effective branching factor (b ) d IDS A*(h 1 ) A*(h 2 )

80 Heuristic quality If n, h 2 (n) h 1 (n) (both admissible) then h 2 dominates h 1 and it is better for search Surely expanded nodes: f n < C h n < C g n If h 2 (n) h 1 (n) then every node expanded for h 2 will also be surely expanded with h 1 (h 1 may also causes some more node expansion) 81

81 More accurate heuristic Max of admissible heuristics is admissible (while it is a more accurate estimate) h n = max(h 1 n, h 2 n ) How about using the actual cost as a heuristic? h(n) = h (n) for all n Will go straight to the goal?! Trade of between accuracy and computation time 82

82 Trivial Heuristics, Dominance Dominance: h a h c if Heuristics form a semi-lattice: Max of admissible heuristics is admissible Trivial heuristics Bottom of lattice is the zero heuristic (what does this give us?) Top of lattice is the exact heuristic

83 Generating heuristics Relaxed problems Inventing admissible heuristics automatically Sub-problems (pattern databases) Learning heuristics from experience 84

84 Relaxed problem Relaxed problem: Problem with fewer restrictions on the actions Optimal solution to the relaxed problem may be computed easily (without search) The cost of an optimal solution to a relaxed problem is an admissible heuristic for the original problem The optimal solution is the shortest path in the super-graph of the statespace. 85

85 Relaxed problem: 8-puzzle 8-Puzzle: move a tile from square A to B if A is adjacent (left, right, above, below) to B and B is blank Relaxed problems 1) can move from A to B if A is adjacent to B (ignore whether or not position is blank) 2) can move from A to B if B is blank (ignore adjacency) 3) can move from A to B (ignore both conditions) Admissible heuristics for original problem (h 1 (n) and h 2 (n)) are optimal path costs for relaxed problems 86 First case: a tile can move to any adjacent square h 2 (n) Third case: a tile can move anywhere h 1 (n)

86 Sub-problem heuristic The cost to solve a sub-problem Store exact solution costs for every possible sub-problem Admissible? The cost of the optimal solution to this problem is a lower bound on the cost of the complete problem 87

87 Pattern databases heuristics Storing the exact solution cost for every possible subproblem instance Combination (taking maximum) of heuristics resulted by different sub-problems Puzzle: 10 3 times reduction in no. of generated nodes vs. h 2

88 Disjoint pattern databases Adding these pattern-database heuristics yields an admissible heuristic?! Dividing up the problem such that each move affects only one sub-problem (disjoint sub-problems) and then adding heuristics 15-puzzle: 10 4 times reduction in no. of generated nodes vs. h 2 24-Puzzle: 10 6 times reduction in no. of generated nodes vs. h 2 Can Rubik s cube be divided up to disjoint sub-problems? 89

89 Learning heuristics from experience Machine Learning Techniques Learn h(n) from samples of optimally solved problems (predicting solution cost for other states) Features of state (instead of raw state description) 8-puzzle number of misplaced tiles number of adjacent pairs of tiles that are not adjacent in the goal state Linear Combination of features 90