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CS 4100: Artificial Intelligence

CS 4100: Artificial Intelligence. Informed Search. Instructor: Jan-Willem van de Meent [Adapted from slides by Dan Klein and Pieter Abbeel for CS188 Intro to AI at UC Berkeley ( ai.berkeley.edu ).]. Announcements. Homework 1: Search (lead TA: Iris)

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CS 4100: Artificial Intelligence

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  1. CS 4100: Artificial Intelligence Informed Search Instructor: Jan-Willem van de Meent [Adapted from slides by Dan Klein and Pieter Abbeel for CS188 Intro to AI at UC Berkeley (ai.berkeley.edu).]

  2. Announcements • Homework 1: Search (lead TA: Iris) • Due Mon 16 Sep at 11:59pm (deadline extended) • Project 1: Search (lead TA: Iris) • Due Mon 23 Sep at 11:59pm • Longer than most – start early! • Homework 2: Constraint Satisfaction Problems (lead TA: Eli) • Due Mon 23 Sep at 11:59pm • Office Hours • Iris: Mon 10.00am-noon, RI 237 • JW: Tue 1.40pm-2.40pm, DG 111 • Eli: Wed 3.00pm-5pm, RY 143 • Zhaoqing: Thu 9.00am-11.00am, HS 202

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

  4. Recap: Search

  5. 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) • Complete: finds solution if it exists • Optimal: finds least-cost plan

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

  7. Example: Pancake Problem

  8. Example: Pancake Problem

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

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

  11. 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

  12. Uninformed Search

  13. 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

  14. UCS in Empty Space

  15. UCS Contours for a Small Maze

  16. Informed Search

  17. 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 10 5 11.2

  18. Example: Heuristic for Travel in Romania h(x)

  19. Example: Heuristic for Pancake Flipping Heuristic: the number of pancakes that is still out of place 2 h(x) 4 3 3 2 0 3 4 3 3 3 2 3

  20. Example: Heuristic for Pancake Flipping New Heuristic: the index of the largest pancake that is still out of place 2 3 h1(x) 4 4 h2(x) 4 3 3 3 3 3 4 2 2 0 0 3 0 3 3 4 4 4 3 3 4 3 3 3 4 3 3 2 2 4 2 3 3 3

  21. Greedy Search

  22. Strategy: Pick node with smallest h(x) h(x)

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

  24. Greedy Search b • Strategy: expand the 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 goal(but finds suboptimal path) • Worst-case: like a badly-guided DFS … b …

  25. Greedy Search in Empty Space

  26. Greedy Search in a Small Maze

  27. A* Search

  28. A* Search UCS(slow and steady) Greedy Search (fast but unreliable) A* Search (best of both worlds)

  29. Combining UCS and Greedy Search • Uniform-cost orders by path cost, or backward cost g(n) • Greedy orders by goal proximity, or forward heuristic 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

  30. 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

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

  32. Admissible Heuristics

  33. 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

  34. Admissible Heuristics • A heuristic h(n) is admissible(optimistic) if: where h*(n) 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

  35. Optimality of A* Tree Search

  36. 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 …

  37. 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 itself!) • 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

  38. 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 itself!) • 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

  39. 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 itself!) • 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 …

  40. Properties of A*

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

  42. 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

  43. A* Search in Empty Space

  44. A* Search in Small Maze

  45. Comparison Greedy Uniform Cost A*

  46. A* Applications • Video games • Pathing / routing problems • Resource planning problems • Robot motion planning • Language analysis • Machine translation • Speech recognition • …

  47. Pacman (Tiny Maze) – UCS / A*

  48. Quiz: Shallow/Deep Water – Guess the 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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