CS 188: Artificial Intelligence Fall 2008

Slides:



Advertisements
Similar presentations
Informed search algorithms
Advertisements

Informed Search CS 171/271 (Chapter 4)
Review: Search problem formulation
Informed Search Algorithms
Informed search strategies
Informed search algorithms
An Introduction to Artificial Intelligence
Solving Problem by Searching
Announcements Homework 1: Search Project 1: Search Sections
CS 343H: Artificial Intelligence
Review: Search problem formulation
CS 188: Artificial Intelligence Fall 2009 Lecture 3: A* Search 9/3/2009 Dan Klein – UC Berkeley Multiple slides from Stuart Russell or Andrew Moore.
CS 460 Spring 2011 Lecture 3 Heuristic Search / Local Search.
CSC344: AI for Games Lecture 4: Informed search
CS 188: Artificial Intelligence Spring 2006 Lecture 2: Queue-Based Search 8/31/2006 Dan Klein – UC Berkeley Many slides over the course adapted from either.
CS 188: Artificial Intelligence Fall 2007 Lecture 6: Robot Motion Planning 9/13/2007 Dan Klein – UC Berkeley Many slides over the course adapted from either.
CS 188: Artificial Intelligence Fall 2006 Lecture 5: Robot Motion Planning 9/14/2006 Dan Klein – UC Berkeley Many slides over the course adapted from either.
Informed search algorithms
Informed search algorithms
Informed search algorithms Chapter 4. Outline Best-first search Greedy best-first search A * search Heuristics.
1 Shanghai Jiao Tong University Informed Search and Exploration.
Quiz 1 : Queue based search  q1a1: Any complete search algorithm must have exponential (or worse) time complexity.  q1a2: DFS requires more space than.
Informed search algorithms Chapter 4. Best-first search Idea: use an evaluation function f(n) for each node –estimate of "desirability"  Expand most.
Informed search strategies Idea: give the algorithm “hints” about the desirability of different states – Use an evaluation function to rank nodes and select.
Review: Tree search Initialize the frontier using the starting state While the frontier is not empty – Choose a frontier node to expand according to search.
Advanced Artificial Intelligence Lecture 2: Search.
For Wednesday Read chapter 6, sections 1-3 Homework: –Chapter 4, exercise 1.
For Wednesday Read chapter 5, sections 1-4 Homework: –Chapter 3, exercise 23. Then do the exercise again, but use greedy heuristic search instead of A*
Princess Nora University Artificial Intelligence Chapter (4) Informed search algorithms 1.
4/11/2005EE562 EE562 ARTIFICIAL INTELLIGENCE FOR ENGINEERS Lecture 4, 4/11/2005 University of Washington, Department of Electrical Engineering Spring 2005.
A General Introduction to Artificial Intelligence.
Feng Zhiyong Tianjin University Fall  Best-first search  Greedy best-first search  A * search  Heuristics  Local search algorithms  Hill-climbing.
Warm-up: Tree Search Solution
CE 473: Artificial Intelligence Autumn 2011 A* Search Luke Zettlemoyer Based on slides from Dan Klein Multiple slides from Stuart Russell or Andrew Moore.
CHAPTER 2 SEARCH HEURISTIC. QUESTION ???? What is Artificial Intelligence? The study of systems that act rationally What does rational mean? Given its.
CS 188: Artificial Intelligence Fall 2006 Lecture 5: CSPs II 9/12/2006 Dan Klein – UC Berkeley Many slides over the course adapted from either Stuart Russell.
CPSC 420 – Artificial Intelligence Texas A & M University Lecture 5 Lecturer: Laurie webster II, M.S.S.E., M.S.E.e., M.S.BME, Ph.D., P.E.
CS 188: Artificial Intelligence Spring 2006 Lecture 5: Robot Motion Planning 1/31/2006 Dan Klein – UC Berkeley Many slides from either Stuart Russell or.
Chapter 3 Solving problems by searching. Search We will consider the problem of designing goal-based agents in observable, deterministic, discrete, known.
Review: Tree search Initialize the frontier using the starting state
Announcements Homework 1 Questions 1-3 posted. Will post remainder of assignment over the weekend; will announce on Slack. No AI Seminar today.
Last time: Problem-Solving
For Monday Chapter 6 Homework: Chapter 3, exercise 7.
Heuristic Search Introduction to Artificial Intelligence
Local Search Algorithms
CS 188: Artificial Intelligence Fall 2007
Artificial Intelligence Problem solving by searching CSC 361
CS 188: Artificial Intelligence Spring 2007
Discussion on Greedy Search and A*
Discussion on Greedy Search and A*
CS 188: Artificial Intelligence Fall 2008
CAP 5636 – Advanced Artificial Intelligence
CS 188: Artificial Intelligence Spring 2007
CS 188: Artificial Intelligence Fall 2008
Lecture 1B: Search.
CS 4100 Artificial Intelligence
Artificial Intelligence Informed Search Algorithms
EA C461 – Artificial Intelligence
Heuristics Local Search
Informed search algorithms
Informed search algorithms
Announcements Homework 1: Search Project 1: Search
More on Search: A* and Optimization
Warm-up: DFS Graph Search
CS 188: Artificial Intelligence Fall 2006
Announcements This Friday Project 1 due Talk by Jeniya Tabassum
Lecture 9 Administration Heuristic search, continued
CS 188: Artificial Intelligence Spring 2006
CS 188: Artificial Intelligence Fall 2007
CS 416 Artificial Intelligence
Presentation transcript:

CS 188: Artificial Intelligence Fall 2008 Lecture 3: A* Search 9/4/2008 Dan Klein – UC Berkeley Many slides over the course adapted from either Stuart Russell or Andrew Moore

Announcements Projects: Pacman Search up today, due 9/12 Work in pairs or alone, hand in one per group 5 slip days for projects, can use up to two per deadline Written Assignments: W1 up, due next week Solve in groups of any size, write up alone One or two questions, handed out in section (and online) Due the next week in section or Thursday lecture We’ll drop the lowest two scores, but no slip days Slides: on web, before lecture

Today A* Search Heuristic Design Local Search

Recap: Search Search problems: Search trees: States (configurations of the world) Successor functions, costs, start and goal tests Search trees: Nodes: represent paths / plans Paths have costs (sum of action costs) Strategies differ (only) in fringe management

Expanding includes incrementing the path cost! General Tree Search Expanding includes incrementing the path cost!

Uniform Cost 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

Best First Strategy: expand nodes which appear closest to goal Heuristic: function which maps states to distance A common case: Best-first takes you straight to the (wrong) goal Worst-case: like a badly-guided DFS b … b …

Example: Heuristic Function h(x)

Combining UCS and Greedy Uniform-cost orders by path cost, or backward cost g(n) Best-first orders by goal proximity, or forward cost h(n) A* Search orders by the sum: f(n) = g(n) + h(n) 5 e h=1 1 1 3 2 S a d G h=5 2 1 h=6 h=2 h=0 1 b c h=5 h=4 Example: Teg Grenager

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

Is A* Optimal? 1 A 3 S G 5 What went wrong? Actual bad goal cost > estimated good goal cost We need estimates to be less than actual costs!

Admissible Heuristics A heuristic h is admissible (optimistic) if: where is the true cost to a nearest goal E.g. Euclidean distance on a map problem Coming up with admissible heuristics is most of what’s involved in using A* in practice.

Optimality of A*: Blocking Proof: What could go wrong? We’d have to have to pop a suboptimal goal G off the fringe before G* This can’t happen: Imagine a suboptimal goal G is on the queue Some node n which is a subpath of G* must be on the fringe (why?) n will be popped before G …

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

UCS vs A* Contours Uniform-cost expanded in all directions A* expands mainly toward the goal, but does hedge its bets to ensure optimality Start Goal Start Goal [demo: position search UCS / A*]

Admissible Heuristics Most of the work is in coming up with admissible heuristics Inadmissible heuristics are often quite effective (especially when you have no choice) Very common hack: use  x h(n) for admissible h,  > 1 to generate a faster but less optimal inadmissible h’ from admissible h

Example: 8 Puzzle What are the states? How many states? What are the actions? What states can I reach from the start state? What should the costs be?

8 Puzzle I Number of tiles misplaced? Why is it admissible? h(start) = This is a relaxed-problem heuristic Average nodes expanded when optimal path has length… …4 steps …8 steps …12 steps ID 112 6,300 3.6 x 106 TILES 13 39 227 8

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 Why admissible? h(start) = Average nodes expanded when optimal path has length… …4 steps …8 steps …12 steps TILES 13 39 227 MAN-HATTAN 12 25 73 3 + 1 + 2 + … = 18

8 Puzzle III How about using the actual cost as a heuristic? Would it be admissible? Would we save on nodes expanded? What’s wrong with it? With A*: a trade-off between quality of estimate and work per node!

Trivial Heuristics, Dominance Dominance: ha ≥ hc if Heuristics form a semi-lattice: Max of admissible heuristics is admissible Trivial heuristics Bottom of lattice is the zero heuristic (what does this give us?) Top of lattice is the exact heuristic

Course Scheduling From the university’s perspective: Set of courses {c1, c2, … cn} Set of room / times {r1, r2, … rn} Each pairing (ck, rm) has a cost wkm What’s the best assignment of courses to rooms? States: list of pairings Actions: add a legal pairing Costs: cost of the new pairing Admissible heuristics? (Who can think of a cs170 answer to this problem?)

Other A* Applications Pathing / routing problems Resource planning problems Robot motion planning Language analysis Machine translation Speech recognition … [demo: plan tiny UCS / A*]

Tree Search: Extra Work? Failure to detect repeated states can cause exponentially more work. Why?

Graph Search In BFS, for example, we shouldn’t bother expanding the circled nodes (why?) S a b d p c e h f r q G

Graph Search Very simple fix: never expand a state twice Can this wreck completeness? Optimality?

Optimality of A* Graph Search Consider what A* does: Expands nodes in increasing total f value (f-contours) Proof idea: optimal goals have lower f value, so get expanded first We’re making a stronger assumption than in the last proof… What?

Consistency Wait, how do we know we expand in increasing f value? Couldn’t we pop some node n, and find its child n’ to have lower f value? YES: What can we require to prevent these inversions? Consistency: Real cost must always exceed reduction in heuristic B h = 0 h = 8 3 g = 10 G A h = 10

Optimality Tree search: Graph search: A* optimal if heuristic is admissible (and non-negative) UCS is a special case (h = 0) Graph search: A* optimal if heuristic is consistent UCS optimal (h = 0 is consistent) In general, natural admissible heuristics tend to be consistent

Summary: A* A* uses both backward costs and (estimates of) forward costs A* is optimal with admissible heuristics Heuristic design is key: often use relaxed problems

Mazeworld Demos

Large Scale Problems with A* What states get expanded? All states with f-cost less than optimal goal cost How far “in every direction” will this be? Intuition: depth grows like the heuristic “gap”: h(start) – g(goal) Gap usually at least linear in problem size Work exponential in depth In huge problems, often A* isn’t enough State space just too big Can’t visit all states with f less than optimal Often, can’t even store the entire fringe Solutions Better heuristics Beam search (limited fringe size) Greedy hill-climbing (fringe size = 1) Goal Start advanced, bonus material

Local Search Methods Queue-based algorithms keep fallback options (backtracking) Local search: improve what you have until you can’t make it better Generally much more efficient (but incomplete)

Types of Problems Planning problems: Identification problems: We want a path to a solution (examples?) Usually want an optimal path Incremental formulations Identification problems: We actually just want to know what the goal is (examples?) Usually want an optimal goal Complete-state formulations Iterative improvement algorithms

Hill Climbing Simple, general idea: Why can this be a terrible idea? Start wherever Always choose the best neighbor If no neighbors have better scores than current, quit Why can this be a terrible idea? Complete? Optimal? What’s good about it?

Hill Climbing Diagram Random restarts? Random sideways steps?

Simulated Annealing Idea: Escape local maxima by allowing downhill moves But make them rarer as time goes on

Simulated Annealing Theoretical guarantee: Stationary distribution: If T decreased slowly enough, will converge to optimal state! Is this an interesting guarantee? Sounds like magic, but reality is reality: The more downhill steps you need to escape, the less likely you are to every make them all in a row People think hard about ridge operators which let you jump around the space in better ways

Beam Search Like greedy search, but keep K states at all times: Variables: beam size, encourage diversity? The best choice in MANY practical settings Complete? Optimal? Why do we still need optimal methods? Greedy Search Beam Search

Genetic Algorithms Genetic algorithms use a natural selection metaphor Like beam search (selection), but also have pairwise crossover operators, with optional mutation Probably the most misunderstood, misapplied (and even maligned) technique around!

Example: N-Queens Why does crossover make sense here? When wouldn’t it make sense? What would mutation be? What would a good fitness function be?

Continuous Problems Placing airports in Romania States: (x1,y1,x2,y2,x3,y3) Cost: sum of squared distances to closest city

Gradient Methods How to deal with continous (therefore infinite) state spaces? Discretization: bucket ranges of values E.g. force integral coordinates Continuous optimization E.g. gradient ascent Image from vias.org