Announcements This Friday Project 1 due Talk by Jeniya Tabassum TweeTIME: A Minimally Supervised Method for Recognizing and Normalizing Time Expressions in Twitter 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
Uniform Cost Search Strategy: expand lowest path cost c1 c2 c3 The good: UCS is complete and optimal! The bad: Explores options in every direction
No information about goal location Start Goal [Demo: contours UCS empty (L3D1)] [Demo: contours UCS pacman small maze (L3D3)] Video of Demo Contours UCS Pacman Small Maze Informed Search 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 Example: Heuristic Function h(x)
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 3 2 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 [Demo: contours greedy empty (L3D1)] [Demo: contours greedy pacman small maze (L3D4)] Video of Demo Contours Greedy (Empty) Video of Demo Contours Greedy (Pacman Small Maze) A*: 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) 8
S g=1 h=5 h=1 e 1 S h=6 c h=7
1 a h=5 1 1 3 b h=6 2
d h=2 G g=2 h=6 A* Search orders by the sum: f(n) = g(n) + h(n) a b
d g=4 h=2 e g=9 h=1 c g=6 G h=0
d g = 10 h=2 h=0 g=3 h=7 g=0 h=6 g = 12 G h=0
Example: Teg Grenager Admissible Heuristics A heuristic h is admissible (optimistic) if: where is the true cost to a nearest goal Examples: 4 15
Coming up with admissible heuristics is most of whats involved in using A* in practice. 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 1. f(n) is less or equal to f(A) 2. f(A) is less than f(B) 3. n expands before B All ancestors of A expand before B A expands before B
A* search is optimal Properties of A* Uniform-Cost b A* b UCS vs A* Contours Uniform-cost expands equally in all
directions Start A* expands mainly toward the goal, but does hedge its bets to ensure optimality Start Goal Goal [Demo: contours UCS / greedy / A* empty (L3D1)]
[Demo: contours A* pacman small maze (L3D5)] Video of Demo Contours (Empty) -- UCS Video of Demo Contours (Empty) -- Greedy Video of Demo Contours (Empty) A* Video of Demo Contours (Pacman Small Maze) A* Comparison Greedy
Uniform Cost A* 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)] Video of Demo Pacman (Tiny Maze) UCS / A* Creating Heuristics
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 366 15 Inadmissible heuristics are often useful too Example: 8 Puzzle Start State
Actions What are the states? How many states? What are the actions? How many successors from the start state? What should the costs be?
Goal State 8 Puzzle I Heuristic: Number of tiles misplaced Why is it admissible? h(start) = 8 This is a relaxed-problem heuristic Start State
Goal State Average nodes expanded when the optimal path has UCS TILES 4 steps 8 steps 12 steps 112 6,300 3.6 x 106 13 39 227
Statistics from Andrew Moore 8 Puzzle II What if we had an easier 8-puzzle where any tile could slide any direction at any time, ignoring other tiles? Total Manhattan distance Start State Goal State Why is it admissible? Average nodes expanded when the optimal path has
8 Puzzle III How about using the actual cost as a heuristic? Would it be admissible? Would we save on nodes expanded? Whats wrong with it? With A*: a trade-off between quality of estimate and work per node As heuristics get closer to the true cost, you will expand fewer nodes but usually do more work per node to compute the heuristic itself Consistency of Heuristics Main idea: estimated heuristic costs actual costs Admissibility: heuristic cost actual cost to goal
A 1 h=4 h=2 h(A) actual cost from A to G C h=1 Consistency: heuristic arc cost actual cost for each arc h(A) h(C) cost(A to C)
3 Consequences of consistency: The f value along a path never decreases G h(A) cost(A to C) + h(C) A* graph search is optimal Optimality of A* Graph Search Sketch: consider what A* does with a consistent heuristic: Fact 1: In tree search, A* expands nodes in
increasing total f value (f-contours) Fact 2: For every state s, nodes that reach s optimally are expanded before nodes that reach s suboptimally Result: A* graph search is optimal f1 f2 f3 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
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