Informed Search

1. Informed Search Russell and Norvig: Ch. 4.1 - 4.3 CSMSC 421 – Fall 2006

2. Outline • Informed = use problem-specific knowledge • Which search strategies? • Best-first search and its variants • Heuristic functions? • How to invent them • Local search and optimization • Hill climbing, simulated annealing, local beam search,…

3. Previously: Graph search algorithm function GRAPH-SEARCH(problem,fringe) return a solution or failure closed an empty set fringe INSERT(MAKE-NODE(INITIAL-STATE[problem]), fringe) loop do if EMPTY?(fringe) then return failure node REMOVE-FIRST(fringe) if GOAL-TEST[problem] applied to STATE[node] succeeds then return SOLUTION(node) if STATE[node] is not in closedthen add STATE[node] to closed fringe INSERT-ALL(EXPAND(node, problem), fringe) A strategy is defined by picking the order of node expansion

4. Blind Search… • …[the ant] knew that a certain arrangement had to be made, but it could not figure out how to make it. It was like a man with a tea-cup in one hand and a sandwich in the other, who wants to light a cigarette with a match. But, where the man would invent the idea of putting down the cup and sandwich—before picking up the cigarette and the match—this ant would have put down the sandwich and picked up the match, then it would have been down with the match and up with the cigarette, then down with the cigarette and up with the sandwich, then down with the cup and up with the cigarette, until finally it had put down the sandwich and picked up the match. It was inclined to rely on a series of accidents to achieve its object. It was patient and did not think… Wart watched the arrangements with a surprise which turned into vexation and then into dislike. He felt like asking why it did not think things out in advance… T.H. White, The Once and Future King

5. Search Algorithms • Blind search – BFS, DFS, uniform cost • no notion concept of the “right direction” • can only recognize goal once it’s achieved • Heuristic search – we have rough idea of how good various states are, and use this knowledge to guide our search

6. Best-first search • General approach of informed search: • Best-first search: node is selected for expansion based on an evaluation function f(n) • Idea: evaluation function measures distance to the goal. • Choose node which appears best • Implementation: • fringe is queue sorted in decreasing order of desirability. • Special cases: Greedy search, A* search

7. Heuristic Webster's Revised Unabridged Dictionary (1913) (web1913) Heuristic \Heu*ris"tic\, a. [Greek. to discover.] Serving to discover or find out. The Free On-line Dictionary of Computing (15Feb98) heuristic 1. <programming> A rule of thumb, simplification or educated guess that reduces or limits the search for solutions in domains that are difficult and poorly understood. Unlike algorithms, heuristics do not guarantee feasible solutions and are often used with no theoretical guarantee. 2. <algorithm> approximation algorithm. From WordNet (r) 1.6 heuristic adj 1: (computer science) relating to or using a heuristic rule 2: of or relating to a general formulation that serves to guide investigation [ant: algorithmic] n : a commonsense rule (or set of rules) intended to increase the probability of solving some problem [syn: heuristic rule, heuristic program]

8. Informed Search • Add domain-specific information to select the best path along which to continue searching • Define a heuristic function, h(n), that estimates the “goodness” of a node n. • Specifically, h(n) = estimated cost (or distance) of minimal cost path from n to a goal state. • The heuristic function is an estimate, based on domain-specific information that is computable from the current state description, of how close we are to a goal

9. Greedy Best-First Search • f(N) = h(N)  greedy best-first

10. Robot Navigation

11. 8 7 6 5 4 3 2 3 4 5 6 7 5 4 3 5 6 3 2 1 0 1 2 4 7 6 5 8 7 6 5 4 3 2 3 4 5 6 Robot Navigation f(N) = h(N), with h(N) = Manhattan distance to the goal

12. Robot Navigation f(N) = h(N), with h(N) = Manhattan distance to the goal 8 7 6 5 4 3 2 3 4 5 6 7 5 4 3 5 0 6 3 2 1 0 1 2 4 What happened??? 7 7 6 5 8 7 6 5 4 3 2 3 4 5 6

13. Greedy Search • f(N) = h(N) Greedy best-first • Is it complete? • Is it optimal? • Time complexity? • Space complexity?

14. More informed search • We kept looking at nodes closer and closer to the goal, but were accumulating costs as we got further from the initial state • Our goal is not to minimize the distance from the current head of our path to the goal, we want to minimize the overall length of the path to the goal! • Let g(N) be the cost of the best path found so far between the initial node and N • f(N) = g(N) + h(N)  A*

15. A* search • Best-known form of best-first search. • Idea: avoid expanding paths that are already expensive. • Evaluation function f(n)=g(n) + h(n) A* • g(n) the cost (so far) to reach the node. • h(n) estimated cost to get from the node to the goal. • f(n) estimated total cost of path through n to goal.

16. 3+8 2+9 4+7 5+6 3+8 8 7 6 5 4 3 2 3 4 5 6 7+2 7 5 4 3 5 6+1 8+3 6+1 6 3 2 1 0 1 2 4 7+2 7+0 6+1 7 6 5 8+1 6+1 8 7 6 5 4 3 2 3 4 5 6 2+9 3+8 7+2 2+9 3+10 4+5 1+10 4+7 5+4 3+6 2+7 8+3 7+4 5+6 6+3 2+9 3+8 4+5 3+6 2+7 6+5 7+4 0+11 1+10 4+7 3+8 5+6 6+3 7+2 5+4 6+3 Robot Navigation f(N) = g(N)+h(N), with h(N) = Manhattan distance to goal 7+0 8+1

17. Can we Prove Anything? • If the state space is finite and we avoid repeated states, the search is complete, but in general is not optimal • If the state space is finite and we do not avoid repeated states, the search is in general not complete • If the state space is infinite, the search is in general not complete

18. Admissible heuristic • Let h*(N) be the true cost of the optimal path from N to a goal node • Heuristic h(N) is admissible if: 0 h(N)  h*(N) • An admissible heuristic is always optimistic

19. Optimality of A*(standard proof) • Suppose suboptimal goal G2 in the queue. • Let n be an unexpanded node on a shortest to optimal goal G. f(G2 ) = g(G2 ) since h(G2 )=0 > g(G) since G2 is suboptimal >= f(n) since h is admissible Since f(G2) > f(n), A* will never select G2 for expansion

20. BUT … graph search • Discards new paths to repeated state. • Previous proof breaks down • Solution: • Add extra bookkeeping i.e. remove more expensive of two paths. • Ensure that optimal path to any repeated state is always first followed. • Extra requirement on h(n): consistency (monotonicity)

21. Consistency • A heuristic is consistent if • If h is consistent, we have i.e. f(n) is nondecreasing along any path.

22. Optimality of A*(more useful) • A* expands nodes in order of increasing f value • Contours can be drawn in state space • Uniform-cost search adds circles. • F-contours are gradually Added: 1) nodes with f(n)<C* 2) Some nodes on the goal Contour (f(n)=C*). Contour I has all Nodes with f=fi, where fi < fi+1.

23. A* search, evaluation • Completeness: YES • Since bands of increasing f are added • Unless there are infinitely many nodes with f<f(G)

24. A* search, evaluation • Completeness: YES • Time complexity: • Number of nodes expanded is still exponential in the length of the solution.

25. A* search, evaluation • Completeness: YES • Time complexity: (exponential with path length) • Space complexity: • It keeps all generated nodes in memory • Hence space is the major problem not time

26. A* search, evaluation • Completeness: YES • Time complexity: (exponential with path length) • Space complexity:(all nodes are stored) • Optimality: YES • Cannot expand fi+1 until fi is finished. • A* expands all nodes with f(n)< C* • A* expands some nodes with f(n)=C* • A* expands no nodes with f(n)>C* Also optimally efficient (not including ties)

27. Memory-bounded heuristic search • Some solutions to A* space problems (maintain completeness and optimality) • Iterative-deepening A* (IDA*) • Here cutoff information is the f-cost (g+h) instead of depth • Recursive best-first search(RBFS) • Recursive algorithm that attempts to mimic standard best-first search with linear space. • (simple) Memory-bounded A* ((S)MA*) • Drop the worst-leaf node when memory is full

28. Recursive best-first search function RECURSIVE-BEST-FIRST-SEARCH(problem) return a solution or failure return RFBS(problem,MAKE-NODE(INITIAL-STATE[problem]),∞) function RFBS( problem, node, f_limit) return a solution or failure and a new f-cost limit if GOAL-TEST[problem](STATE[node]) then return node successors  EXPAND(node, problem) ifsuccessors is empty then return failure, ∞ for eachsinsuccessorsdo f [s]  max(g(s) + h(s), f [node]) repeat best the lowest f-value node in successors iff [best] > f_limitthen return failure, f [best] alternative the second lowest f-value among successors result, f [best]  RBFS(problem, best, min(f_limit, alternative)) ifresult failure then returnresult

29. Recursive best-first search • Keeps track of the f-value of the best-alternative path available. • If current f-values exceeds this alternative f-value than backtrack to alternative path. • Upon backtracking change f-value to best f-value of its children. • Re-expansion of this result is thus still possible.

30. RBFS evaluation • RBFS is a bit more efficient than IDA* • Still excessive node generation (mind changes) • Like A*, optimal if h(n) is admissible • Space complexity is O(bd). • IDA* retains only one single number (the current f-cost limit) • Time complexity difficult to characterize • Depends on accuracy if h(n) and how often best path changes. • IDA* and RBFS suffer from too little memory.

31. (simplified) memory-bounded A* • Use all available memory. • I.e. expand best leafs until available memory is full • When full, SMA* drops worst leaf node (highest f-value) • Like RFBS backup forgotten node to its parent • What if all leafs have the same f-value? • Same node could be selected for expansion and deletion. • SMA* solves this by expanding newest best leaf and deleting oldest worst leaf. • SMA* is complete if solution is reachable, optimal if optimal solution is reachable.

32. Heuristic functions • E.g for the 8-puzzle • Avg. solution cost is about 22 steps (branching factor +/- 3) • Exhaustive search to depth 22: 3.1 x 1010 states. • A good heuristic function can reduce the search process.

33. 1 2 3 4 5 6 7 8 5 8 4 2 1 7 3 6 Heuristic Function • Function h(N) that estimates the cost of the cheapest path from node N to goal node. • Example: 8-puzzle h1(N) = number of misplaced tiles = 6 goal N

34. 1 2 3 4 5 6 7 8 5 8 4 2 1 7 3 6 Heuristic Function • Function h(N) that estimate the cost of the cheapest path from node N to goal node. • Example: 8-puzzle h2(N) = sum of the distances of every tile to its goal position = 2 + 3 + 0 + 1 + 3 + 0 + 3 + 1 = 13 goal N

35. 3 3 4 5 3 4 2 4 2 1 3 3 0 4 5 4 8-Puzzle f(N) = h1(N) = number of misplaced tiles

36. 3+3 1+5 2+3 3+4 5+2 0+4 3+2 4+1 2+3 1+3 5+0 3+4 1+5 2+4 8-Puzzle f(N) = g(N) + h(N) with h1(N) = number of misplaced tiles

37. 6 5 2 5 2 1 3 4 0 4 6 5 8-Puzzle f(N) = h2(N) =  distances of tiles to goal

38. 0+4 8-Puzzle EXERCISE: f(N) = g(N) + h2(N) with h2(N) =  distances of tiles to goal

39. Heuristic quality • Effective branching factor b* • Is the branching factor that a uniform tree of depth d would have in order to contain N+1 nodes. • Measure is fairly constant for sufficiently hard problems. • Can thus provide a good guide to the heuristic’s overall usefulness. • A good value of b* is 1.

40. Heuristic quality and dominance • 1200 random problems with solution lengths from 2 to 24. • If h2(n) >= h1(n) for all n (both admissible) then h2 dominates h1 and is better for search

41. Inventing admissible heuristics • Admissible heuristics can be derived from the exact solution cost of a relaxed version of the problem: • Relaxed 8-puzzle for h1 : a tile can move anywhere As a result, h1(n) gives the shortest solution • Relaxed 8-puzzle for h2 : a tile can move to any adjacent square. As a result, h2(n) gives the shortest solution. The optimal solution cost of a relaxed problem is no greater than the optimal solution cost of the real problem.

42. Another approach… Local Search • Previously: systematic exploration of search space. • Path to goal is solution to problem • YET, for some problems path is irrelevant. • E.g 8-queens • Different algorithms can be used:Local Search • Hill-climbing or Gradient descent • Simulated Annealing • Genetic Algorithms, others… • Also applicable to optimization problems • systematic search doesn’t work • however, can start with a suboptimal solution and improve it

43. Local search and optimization • Local search= use single current state and move to neighboring states. • Advantages: • Use very little memory • Find often reasonable solutions in large or infinite state spaces. • Are also useful for pure optimization problems. • Find best state according to some objective function.

44. Local search and optimization

45. Hill-climbing search • “is a loop that continuously moves in the direction of increasing value” • It terminates when a peak is reached. • Hill climbing does not look ahead of the immediate neighbors of the current state. • Hill-climbing chooses randomly among the set of best successors, if there is more than one. • Hill-climbing a.k.a. greedy local search • Some problem spaces are great for hill climbing and others are terrible.

46. Hill-climbing search function HILL-CLIMBING(problem) return a state that is a local maximum input:problem, a problem local variables: current, a node. neighbor, a node. current  MAKE-NODE(INITIAL-STATE[problem]) loop do neighbor a highest valued successor of current if VALUE[neighbor] ≤ VALUE[current] then return STATE[current] current neighbor

47. Robot Navigation Local-minimum problem f(N) = h(N) = straight distance to the goal

48. Examples of problems with HC • applet

49. Drawbacks of hill climbing • Problems: • Local Maxima: peaks that aren’t the highest point in the space • Plateaus: the space has a broad flat region that gives the search algorithm no direction (random walk) • Ridges: flat like a plateau, but with dropoffs to the sides; steps to the North, East, South and West may go down, but a combination of two steps (e.g. N, W) may go up. • Remedy: • Introduce randomness

50. Hill-climbing variations • Stochastic hill-climbing • Random selection among the uphill moves. • The selection probability can vary with the steepness of the uphill move. • First-choice hill-climbing • Stochastic hill climbing by generating successors randomly until a better one is found. • Random-restart hill-climbing • Tries to avoid getting stuck in local maxima. • If at first you don’t succeed, try, try again…