Informed Search A* Algorithm CE417: Introduction to Artificial Intelligence Sharif University of Technology Spring 2019 Soleymani “Artificial Intelligence: A Modern Approach”, Chapter 3 Most slides have been adopted from Klein and Abdeel, CS188, UC Berkeley.
Outline } Heuristics } Greedy (best-first) search } A * search } Finding heuristics 2
Uninformed Search 3
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 4
UCS Example 5
Informed Search 6
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 10 5 11.2 7
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 𝑜 : estimated cost of cheapest path from 𝑜 to a goal } Depends only on 𝑜 (not path from root to 𝑜 ) } If 𝑜 is a goal state then ℎ(𝑜) =0 } ℎ(𝑜) ≥ 0 } Examples of heuristic functions include using a rule-of-thumb, an educated guess, or an intuitive judgment 8
Example: Heuristic Function h(x) 9
Greedy Search 10
Greedy search } Priority queue based on ℎ(𝑜) } e.g., ℎ '() 𝑜 = straight-line distance from n to Bucharest } Greedy search expands the node that appears to be closest to goal Greedy 11
Example: Heuristic Function h(x) 12
Romania with step costs in km 13
Greedy best-first search example 14
Greedy best-first search example 15
Greedy best-first search example 16
Greedy best-first search example 17
Greedy Search } Expand the node that seems closest… } What can go wrong? 18
Greedy Search } Strategy: expand a node that you think is closest to a goal state b … Heuristic: estimate of distance to nearest goal for each state } } A common case: Best-first takes you straight to the (wrong) goal } b … } Worst-case: like a badly-guided DFS 19
Properties of greedy best-first search } Complete? No } Similar to DFS, only graph search version is complete in finite spaces } Infinite loops, e.g., (Iasi to Fagaras) Iasi à Neamt à Iasi à Neamt } Time } 𝑃(𝑐 𝑛 ) , but a good heuristic can give dramatic improvement } Space } 𝑃(𝑐 𝑛 ) : keeps all nodes in memory } Optimal? No 20
Greedy Search 21
A* Search 22
A * search } Idea: minimizing the total estimated solution cost } Evaluation function for priority 𝑔 𝑜 = 𝑜 + ℎ(𝑜) } 𝑜 = cost so far to reach 𝑜 } ℎ 𝑜 = estimated cost of the cheapest path from 𝑜 to goal } So, 𝑔 𝑜 = estimated total cost of path through 𝑜 to goal Actual cost 𝑜 Estimated cost ℎ 𝑜 … … start n goal 𝑔 𝑜 = 𝑜 + ℎ(𝑜) 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) } g = 0 8 S h=6 g = 1 h=1 e a h=5 1 1 3 2 g = 9 g = 2 g = 4 S a d G b d e h=1 h=6 h=2 h=6 h=5 1 h=2 h=0 1 g = 3 g = 6 g = 10 c b c G d h=7 h=0 h=2 h=7 h=6 g = 12 G h=0 A* Search orders by the sum: f(n) = g(n) + h(n) } Example: Teg Grenager 25
When should A* terminate? } Should we stop when we enqueue a goal? h = 2 A 2 2 S G h = 3 h = 0 2 3 B h = 1 } No: only stop when we dequeue a goal 26
Is A* Optimal? h = 6 1 3 A 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 Admissible (optimistic) heuristics slow down optimality by trapping good plans on the frontier 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 } f(n) is less or equal to f(A) 1. Definition of f-cost 𝑔 𝑜 ≤ 𝑜 + ℎ ∗ (𝑜) Admissibility of h 𝐵 = 𝑜 + ℎ ∗ (𝑜) 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 } f(n) is less or equal to f(A) 1. f(A) is less than f(B) 2. 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 } f(n) is less or equal to f(A) 1. f(A) is less than f(B) 2. n expands before B 3. 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(𝑜) has some properties 36
A * search: example 37
A * search: example 38
A * search: example 39
A * search: example 40
A * search: example 41
A * search: example 42
Graph Search 43
Tree Search: Extra Work! } Failure to detect repeated states can cause exponentially more work. State Graph Search Tree 44
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? } 46
Optimality of A* Graph Search 47
A* Graph Search Gone Wrong? State space graph Search tree A S (0+2) 1 1 h=4 S C A (1+4) B (1+1) 1 h=1 h=2 2 3 C (2+1) C (3+1) B h=1 G (5+0) G (6+0) G h=0 48
Conditions for optimality of A * } Admissibility: ℎ(𝑜) be a lower bound on the cost to reach goal } Condition for optimality of TREE-SEARCH version of A * } Consistency (monotonicity): ℎ 𝑜 ≤ 𝑑 𝑜, 𝑏, 𝑜 8 + ℎ 𝑜 8 } Condition for optimality of GRAPH-SEARCH version of A * 49
Consistent heuristics } Triangle inequality for every node 𝑜 and every successor 𝑜 8 generated by any action 𝑏 𝑜 ℎ(𝑜) 𝑑(𝑜, 𝑏, 𝑜′) ℎ 𝑜 ≤ 𝑑 𝑜, 𝑏, 𝑜 8 + ℎ 𝑜 8 𝐻 ℎ(𝑜′) 𝑜′ 𝑑 𝑜, 𝑏, 𝑜 8 : cost of generating 𝑜′ by applying action to 𝑜 50
Consistency of Heuristics } Main idea: estimated heuristic costs ≤ actual costs Admissibility: heuristic cost ≤ actual cost to goal } h(A) ≤ actual cost from A to G A Consistency: heuristic “arc” cost ≤ actual cost for each arc } 1 h(A) – h(C) ≤ cost(A to C) C h=4 h=1 h=2 } Consequences of consistency: 3 The f value along a path never decreases } h(A) ≤ cost(A to C) + h(C) A* graph search is optimal } G 51
Admissible but not consistent: Example 𝑜 = 5 𝑜 ℎ 𝑜 = 9 𝑑(𝑜, 𝑏, 𝑜’) = 1 𝑔(𝑜) = 14 ℎ(𝑜) = 9 1 ℎ(𝑜’) = 6 10 ⟹ ℎ 𝑜 ≰ ℎ 𝑜’ + 𝑑(𝑜, 𝑏, 𝑜’) 𝑜′ = 6 ℎ 𝑜 8 = 6 𝑜′ 𝑔(𝑜′) = 12 10 G } 𝑔 (for admissible heuristic) may decrease along a path } Is there any way to make ℎ consistent? (ℎ 𝑜 8 , ℎ D 𝑜′ = max D 𝑜 − 𝑑(𝑜, 𝑏, 𝑜′)) ℎ 52
Consistency implies admissibility } Consistency ⇒ Admissblity } All consistent heuristic functions are admissible } Nonetheless, most admissible heuristics are also consistent 𝑑(𝑜 J , 𝑏 J , 𝑜 K ) 𝑑(𝑜 K , 𝑏 K , 𝑜 L ) 𝑑(𝑜 P , 𝑏 P , 𝐻) 𝑜 J 𝑜 K 𝑜 L 𝑜 P 𝐻 … ℎ 𝑜 J ≤ 𝑑 𝑜 J , 𝑏 J , 𝑜 K + ℎ ( 𝑜 K ) ≤ 𝑑 𝑜 J , 𝑏 J , 𝑜 K + 𝑑 𝑜 K , 𝑏 K , 𝑜 L + ℎ ( 𝑜 L ) … P ≤ ∑ 𝑑 𝑜 N , 𝑏 N , 𝑜 NOJ + ℎ(G) ⇒ ℎ 𝑜 J ≤ cost of (every) path from 𝑜 J to goal 0 NQJ ≤ cost of optimal path from 𝑜 J to goal 53
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 54
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