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CS 188: Artificial Intelligence

CS 188: Artificial Intelligence. Informed Search. Instructors: Dan Klein and Pieter Abbeel University of California, Berkeley [These slides were created by Dan Klein and Pieter Abbeel for CS188 Intro to AI at UC Berkeley. All CS188 materials are available at http:// ai.berkeley.edu .]. Today.

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CS 188: Artificial Intelligence

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  1. CS 188: Artificial Intelligence Informed Search Instructors: Dan Klein and Pieter Abbeel University of California, Berkeley [These slides were created by Dan Klein and Pieter Abbeel for CS188 Intro to AI at UC Berkeley. All CS188 materials are available at http://ai.berkeley.edu.]

  2. Today • Informed Search • Heuristics • Greedy Search • A* Search • Graph Search

  3. Recap: Search

  4. Recap: Search • Search problem: • States (configurations of the world) • Actions and costs • Successor function (world dynamics) • Start state and goal test • Search tree: • Nodes: represent plans for reaching states • Plans have costs (sum of action costs) • Search algorithm: • Systematically builds a search tree • Chooses an ordering of the fringe (unexplored nodes) • Optimal: finds least-cost plans

  5. Example: Pancake Problem Cost: Number of pancakes flipped

  6. Example: Pancake Problem

  7. Example: Pancake Problem State space graph with costs as weights 4 2 3 2 3 4 3 4 2 3 2 2 4 3

  8. General Tree Search Path to reach goal: Flip four, flip three Total cost: 7 Action: flip top twoCost: 2 Action: flip all fourCost: 4

  9. The One Queue • All these search algorithms are the same except for fringe strategies • Conceptually, all fringes are priority queues (i.e. collections of nodes with attached priorities) • Practically, for DFS and BFS, you can avoid the log(n) overhead from an actual priority queue, by using stacks and queues • Can even code one implementation that takes a variable queuing object

  10. Uninformed Search

  11. Uniform Cost Search • Strategy: expand lowest path cost • The good: UCS is complete and optimal! • The bad: • Explores options in every “direction” • No information about goal location c  1 … c  2 c  3 Start Goal [Demo: contours UCS empty (L3D1)] [Demo: contours UCS pacmansmall maze (L3D3)]

  12. Video of Demo Contours UCS Empty

  13. Video of Demo Contours UCS Pacman Small Maze

  14. Informed Search

  15. Search Heuristics • A heuristic is: • A function that estimateshow close a state is to a goal • Designed for a particular search problem • Examples: Manhattan distance, Euclidean distance for pathing 10 5 11.2

  16. Example: Heuristic Function h(x)

  17. Example: Heuristic Function Heuristic: the number of the largest pancake that is still out of place 3 h(x) 4 3 4 3 0 4 4 3 4 4 2 3

  18. Greedy Search

  19. Example: Heuristic Function h(x)

  20. Greedy Search • Expand the node that seems closest… • What can go wrong?

  21. Greedy Search b • Strategy: expand a node that you think is closest to a goal state • Heuristic: estimate of distance to nearest goal for each state • A common case: • Best-first takes you straight to the (wrong) goal • Worst-case: like a badly-guided DFS … b … [Demo: contours greedy empty (L3D1)] [Demo: contours greedy pacman small maze (L3D4)]

  22. Video of Demo Contours Greedy (Empty)

  23. Video of Demo Contours Greedy (Pacman Small Maze)

  24. A* Search

  25. A* Search UCS Greedy A*

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

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

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

  29. Admissible Heuristics

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

  31. Admissible Heuristics • A heuristic h is admissible(optimistic) if: where is the true cost to a nearest goal • Examples: • Coming up with admissible heuristics is most of what’s involved in using A* in practice. 15 4

  32. Optimality of A* Tree Search

  33. 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 fringe before B …

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

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

  36. Optimality of A* Tree Search: Blocking Proof: • Imagine B is on the fringe • Some ancestor n of A is on the fringe, too (maybe A!) • Claim: n will be expanded before B • f(n) is less or equal to f(A) • f(A) is less than f(B) • n expands before B • All ancestors of A expand before B • A expands before B • A* search is optimal …

  37. Properties of A*

  38. Properties of A* Uniform-Cost A* b b … …

  39. UCS vs A* Contours • Uniform-cost expands equally in all “directions” • A* expands mainly toward the goal, but does hedge its bets to ensure optimality Start Goal Goal Start [Demo: contours UCS / greedy / A* empty (L3D1)] [Demo: contours A* pacmansmall maze (L3D5)]

  40. Video of Demo Contours (Empty) -- UCS

  41. Video of Demo Contours (Empty) -- Greedy

  42. Video of Demo Contours (Empty) – A*

  43. Video of Demo Contours (Pacman Small Maze) – A*

  44. Comparison Greedy Uniform Cost A*

  45. A* Applications

  46. A* Applications • Video games • Pathing / routing problems • Resource planning problems • Robot motion planning • Language analysis • Machine translation • Speech recognition • … [Demo: UCS / A* pacman tiny maze (L3D6,L3D7)] [Demo: guess algorithm Empty Shallow/Deep (L3D8)]

  47. Video of Demo Pacman (Tiny Maze) – UCS / A*

  48. Video of Demo Empty Water Shallow/Deep – Guess Algorithm

  49. Creating Heuristics

  50. 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 366 15

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