ARTIFICIAL INTELLIGENCE. Informed search
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1 INFOB2KI Utrecht University The Netherlands ARTIFICIAL INTELLIGENCE Informed search Lecturer: Silja Renooij These slides are part of the INFOB2KI Course Notes available from
2 Shakey ( ) Shakey is a robot which navigates using? a) Dijkstra s algorithm b) A1 c) A2 d) A* 2
3 Recap: Search Search problem: States (configurations of the world) Actions and costs Successor function Start state and goal test Search tree: Nodes represent how to reach states Cost of reaching state = sum of action costs Search algorithm: Systematically builds a search tree Chooses an ordering of the fringe Optimal: find least cost plans 3
4 Search Heuristics A heuristic is: a function that estimates how close a state is to a goal designed for a particular search problem Examples: Manhatten distance Euclidean distance easy to compute: otherwise overhead of computing the heuristic could outweigh time saved by reducing search! 4
5 Heuristic search (outline) Best first search Greedy search A * search Heuristic functions Local search algorithms Hill climbing search Simulated annealing search Local beam search Genetic algorithms 5
6 Best-first search Tree/Graph search with an evaluation function f(n) for each node n estimate of "desirability" Expand most desirable unexpanded node Should direct search toward goal Implementation: Order the nodes in fringe in decreasing order of desirability f(n) Finds best solution, according to evaluation function Special cases: greedy search A * search Both also use heuristic h(n) with h(n) 0 for all n h(n) = 0 for state[n] = goal 6
7 Romania with step costs in km and straight line distances between city map coordinates 7
8 Greedy search Evaluation function f(n) = h(n) where heuristic h(n) = estimated cost from n to goal e.g., h SLD (n) = straight line distance from n to Bucharest Note: h SLD (n) 0 for all n; h SLD (Bucharest) = 0 Greedy best first search expands the node that appears to be closest to goal Goal test: upon generation = upon expansion 8
9 Greedy search example f(arad) = h sld (Arad) = 366 Expand Arad 9
10 Greedy search example 10
11 Greedy search example 11
12 Greedy search example Done, or continue? Resembles DFS? 12
13 Properties of greedy search Complete? TREE SEARCH: No; 1 GRAPH SEARCH: Yes (in finite state spaces) Optimal? No Time? O(b m ) (but a good heuristic can give dramatic improvement) Space? O(b m ) (keeps all nodes in memory) Properties similar to DFS without space benefit; behaviour not necessarily 1 consider e.g. finding a path from Neamt to Fagaras with SLD heuristic, then Iasi Neamt Iasi Neamt since from Iasi neighbour Neamt is closer to Fagaras than neighbour Vaslui 13
14 A * search Combines benefits of greedy best first search and uniform cost search Evaluation function f(n) = g(n) + h(n) where g(n) = cost so far to reach n (= path cost; dynamic) h(n) = estimated cost from n to goal (= static heuristic) f(n) = estimated total cost of path through n to goal A * search avoids expanding paths that are already expensive Goal test upon expansion! 14
15 A * search example f(arad) = g(arad) + h sld (Arad) = Expand Arad 15
16 A * search example min g(n) Expand n with minimal f(n) 16
17 A * search example min g(n) Expand n with minimal f(n) 17
18 A * search example min g(n) Expand n with minimal f(n) 18
19 A * search example min g(n) Done, or continue? For Bucharest: f(n) <> min f(n) g(n) <> min g(n) h(n) = 0 19
20 A * search example min g(n) Done, or continue? For Bucharest: f(n) = min f(n) g(n) <> min g(n) h(n) = 0 A * vs Dijkstra: 20
21 Admissible heuristics Definition: a heuristic h(n) is admissible if h(n) h * (n) for every node n where h * (n) is the optimal heuristic, i.e. gives true cost to reach the goal state from n. An admissible heuristic never overestimates the actual cost to reach the goal, i.e., it is optimistic (property transfers to f(n)) Example: h SLD (n) (never overestimates the actual road distance) Theorem: If h(n) is admissible then A * using TREE SEARCH is optimal 21
22 Optimality of A * (proof tree) Consider: Optimal goal G A fringe with (generated, but not yet expanded): suboptimal goal G 2 noden, on a shortest path to G To prove: G 2 will never be expanded before n. First we show that f(g 2 ) > f(g): 1) f(g) = g(g) since h(g) = 0 2) f(g 2 ) = g(g 2 ) since h(g 2 ) = 0 3) g(g 2 ) > g(g) G 2 suboptimal, so higher (path) costs f(g 2 ) > f(g) from 1 3) 22
23 Optimality of A * (proof-cntd) Recall: G optimal; fringe with suboptimal G 2 and n on shortest path to G To prove: G 2 will never be expanded before n. Established: f(g 2 ) > f(g) Next we show that f(g 2 ) > f(n): 1) h(n) h*(n) assumption h is admissible h(n) + g(n) h * (n) + g(n) add g(n) to both sides f(n) g(g) n on shortest path to G; def f 2) f(g) = g(g) + 0 h(g) = 0; def f f(g) f(n) 1,2 Hence f(g 2 ) > f(n), and A * will never select G 2 for expansion 23
24 Consistent heuristics A heuristic is consistent (or monotonic) if for every node n and every successor n' of n, generated by any action a : h(n) c(n,a,n') + h(n') where c(n,a,n') is the step cost If h is consistent, we have f(n') = g(n') + h(n') (def) = g(n) + c(n,a,n') + h(n') g(n) + h(n) (consistent) = f(n) (def) i.e., f(n) is non decreasing along any path. Intuition: h should be optimistic, but not to the extent that total costs f can drop by taking a next step 24
25 Optimality of A * (again) Theorem: If h(n) is consistent, A* using GRAPH SEARCH is optimal A * expands nodes in order of increasing f value Gradually adds "f contours" of nodes Contour f c has all nodes n with f(n) c Contours f 380 f 400 f
26 Properties of A* Complete? Yes (unless there are infinitely many nodes with f f(g) = f * ) Optimal? Yes (if heuristic is consistent (GRAPH) /admissible (TREE) ) Time? Exponential in d Space? Keeps all nodes in memory 26
27 Admissible heuristics Recall the properties of (admissible) heuristic h: h(goal node) = 0 0 h(n) h*(n) for every node n (admissible: should never overestimate!) E.g., two common heuristics for the 8 puzzle: h 1 (n) = number of misplaced tiles h 2 (n) = total Manhattan distance (i.e., no. of squares from desired location of each tile (only horizontal and vertical moves!!))) h 1 (Start) =? h 2 (Start) =? 27
28 Admissible heuristics E.g., two common heuristics for the 8 puzzle: h 1 (n) = number of misplaced tiles h 2 (n) = total Manhattan distance h 1 (Start) = 8 h 2 (Start) = = 18 28
29 Which is better? - Dominance If h 2 (n) h 1 (n) for all n (and both admissible) then h 2 dominates h 1 h 2 is better for search Typical search costs (average number of nodes expanded over 100 instances of 8 puzzle per solution length d ): d=12 d=24 IDS = 3,644,035 nodes A * (h 1 ) = 227 nodes A * (h 2 ) = 73 nodes IDS = too many nodes A * (h 1 ) = 39,135 nodes A * (h 2 ) = 1,641 nodes 29
30 Good heuristics h*(n) h(n) 0 for all n (h is admissible) The closer h is to h*, the less nodes are kept open for search (more efficient) If h(n)=h(n)* for all n then A* leads direct to optimal solution without search If h(n)=0 for all n then A* is uniform cost search (= Dijkstra s algorithm + goal test) However, if h is not admissible then optimal solution can be missed! 30
31 Relaxed problems Relaxed problem: aproblem with fewer restrictions on the actions E.g. allow moves to occupied slots in 8 puzzle The cost of an optimal solution to a relaxed problem is an admissible heuristic for the original problem If the rules of the 8 puzzle are further relaxed so that a tile can move anywhere in one step h 1 (n) gives the best (shortest) solution a tile can move to any adjacent square h 2 (n) gives the shortest solution 31
32 Variations on A* Hierarchical A* IDA* (Iterative Deepening A*) SMA* (Simplified Memory bounded A*) Specific for dynamic environments: LPA* (Lifelong Planning A*) D* (A* for dynamic goals) 32
33 Summary best-first search Similar to uninformed search: tree/graph search goal test step & path costs solution = (least cost) path through statespace more problem specific ingredients (beyond definition of problem): Heuristic function h(n) Evaluation function f(n) 33
34 Local search algorithms In many optimization problems, the path to the goal is irrelevant; the goal state itself is the solution State space = set of "complete state configurations (rather than partial/incremental) Search: find configuration satisfying constraints e.g. n queens In such cases, we can use local search algorithms keep a single "current" state, and try to improve it by moving to neighboring states find goal state, or state that optimizes some objective function (an evaluation function) 34
35 Example: TSP B C A E D Which is the shortest route that visits all cities exactly once, and returns at the starting point? 35
36 Hill-climbing search No search tree Terminates when it reaches peak (steepest ascent version) No look ahead beyond immediate neighbors (aka greedy local search) # VALUE = value from objective function "Like climbing Everest in thick fog with amnesia" 36
37 Hill-climbing search Problem: depending on initial state, can easily get stuck in local maxima 37
38 Hill-climbing: 8-queens problem h = # of pairs of queens that attack each other, either directly or indirectly h = 17 for the above state squares show h(neighbors) Objective: h=0 (minimization: gradient descent) 38
39 Hill-climbing: 8-queens problem A local minimum with h = 1 (every neighbor has higher cost) 39
40 Simulated annealing search Combines hill climbing with random walk Idea: escape local max/min by allowing some random" moves ( shake ) but gradually decrease their frequency and intensity 40
41 Simulated annealing search # T decreases over time! # instead of best # go if better # possibly try worse anyway Property: if T decreases slowly enough, a global optimum is found with probability approaching 1 Widely used in VLSI layout, airline scheduling, etc 41
42 Local beam search Keep track of k states in memory rather than just one Start with k randomly generated states At each iteration, all the successors of all k states are generated If any one is a goal state, stop; else select the k best successors from the complete list and repeat. Come on over here, the grass is greener! 42
43 Genetic algorithms A successor state is generated by combining two parent states Start with k randomly generated states (population) A state (or, individual) is represented as a string over a finite alphabet (often a string of 0s and 1s) Objective function (or, fitness function): higher values for better states. Produce the next generation of states by selection, crossover, and mutation 43
44 Summary Informed Search Use domain knowledge to guide search towards goal. If path to goal is relevant: best first search If only finding the goal is relevant: local search What about?: search vsoptimization vslearning 44
45 Shakey ( ) Shakey is a robot which navigates using? a) Dijkstra s algorithm b) A1 c) A2 d) A* 45
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