Chapter04 Beyond Classical Search - csie.ntu.edu.tw

Chapter 4

Beyond Classical Search

Jane Hsu

National Taiwan University

Acknowledgements: This presentation is created by Jane hsu based on the lecture slides from The Artificial Intelligence: A Modern Approach by Russell & Norvig, a PowerPoint version by Min-Yen Kan, as well as various materials from the web.

Beyond Classical Search © Jane Hsu 2

Outline

� Local Search Algorithms

� Hill climbing

� Simulated annealing

� Genetic algorithms

� Local search in continuous spaces

� Searching with nondeterministic actions

� Searching with partial observations

� Online Search

10/02/2012 © Jane Hsu 3

General Search Problem

� Given:

� Problem space (or state space)

� A set of nodes N (each representing a problem state)

� A successor function Next(n) defining the next states

� Start node

� Goal: a subset of N

� To find

� One path from the start node to a goal node

� All paths from the start node to any goal node

� The best path from the start to the best goal


State Representation

� A state is a “black box” – any data

structure that supports

� successor function

� heuristic function, and

� goal test

Mutilated Checker Board

10/02/2012 © Jane Hsu 5


Iterative Improvement Algorithms

� In many optimization problems, path is irrelevant; the goal state itself is the solution

� State space = set of "complete" configurations

� Find optimal configuration, e.g., TSP

� Find configuration satisfying constraints, e.g., n-queens

� In such cases, we can use iterative improvement, also called local search algorithms

� keep a single "current" state, try to improve it

� Constant space

� Online or offline search


Traveling Salesperson Problem

� Start with any complete tour, perform pair-wise exchanges.

� Variants of this approach get within 1% of optimal very quickly with thousands of cities.

Local Search Algorithms

Jane Hsu

National Taiwan University



Hill-Climbing (Gradient Ascent/Descent)

“Like climbing Everest in thick fog with amnesia”

Hill Climbing


Local Maxima



Problems in Hill-Climbing Search

� Consider the state space landscape


Simulated Annealing

� Idea: escape local maxima by allowing some "bad" moves

but gradually decrease their size & frequency

�


Simulated Annealing

� Idea: escape local maxima by allowing some "bad" moves

but gradually decrease their size & frequency

�


Properties of Simulated Annealing

� At fixed temperature T, state occupation probability reaches Boltzman distribution

� One can prove: If T decreases slowly enough, then simulated annealing search will find a global optimum with probability approaching 1

� Devised by Metropolis et al. (1953) for physical process modeling

� Widely used in VLSI layout, airline scheduling, etc

kT

xE

exP

)(

)(

∆

= α

Properties of Simulated Annealing



Local Beam Search

� Keep track of (top) k states rather than just one� Start with k randomly generated states

� At each iteration, all the successors of all k states are generated

� If any one is a goal state, stop; else select the k best successors from the complete list and repeat.

� Question: is it k searches run in parallel?

� Problem: all k states end up on same local hill

� Solution idea: choose k successors randomly, biased towards good ones.


Genetic Algorithms

� Population: Start with k randomly generated individuals (i.e. states)

� Individual: each is represented as a string over a finite alphabet (often a string of 0s and 1s)

� Fitness function: evaluation of the “goodness”

of a given state.

� A successor is generated by combining two parents from the current population.

� Produce the next generation of states by selection, crossover, and mutation


Genetic Algorithms

� Fitness function: number of non-attacking pairs of queens (min = 0, max = 8 × 7/2 = 28)

� 24/(24+23+20+11) = 31%

� 23/(24+23+20+11) = 29% etc


Crossover Operators

� States are encoded as strings


Local Search in

Continuous Spaces


Example: Airport Site Planning

� Suppose we want to place three new airports anywhere in Romania, such all cities have easy access to airports.

� 6-D state space (x1,y1),(x2,y2),(x3,y3)

� Objective function

f(x1,y1,x2,y2,x3,y3) = sum of squared distances from each city to its nearest airport


Continuous Search Spaces

� Most real-world environments are continuous.

� Successor function would return infinitely many states!

� Solution ideas

� Discretization

� Gradient of the objective function

� Empirical gradient

� Line search

� Newton-Raphson method to solve

Continuous Search Space


Searching with

Nondeterministic Actions

Possible States of Vacuum World


Erratic Vacuum World

� Action: Suck

� When applied to a dirty square the action cleans the square and sometimes cleans up dirt in an adjacent square, too.

� When applied to a clean square the action sometimes deposit dirt on the carpet.

� Use a Results function that returns a set of possible outcome states.

� Solution?


Contingency Plan

� [Suck, if State=5 then [Right, Such] else []]

� Solutions for nondeterministic problems can contain nested if-then-else statements.

� Selection of actions based on contingencies arising during execution.

� Exact prediction is impossible for many real-world problems.

� People keep their eyes open while walking/driving


AND-OR Search Trees


AND-OR Graph Search


AND-OR Graph Search


Slippery Vacuum World

� Original vacuum world +

� Movement actions sometimes fail, leaving the agent in the same location.

� There is no way to move reliably.

� All solutions for this problem are cyclic plans


Search Graph with Cycles


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Chapter04 Beyond Classical Search - csie.ntu.edu.tw