Local Search Pat Riddle 2012 Semester 2 Patricia J Riddle Adapted from slides by Stuart Russell, .

Local SearchPat Riddle

2012 Semester 2Patricia J Riddle

Adapted from slides by Stuart Russell, http://aima.cs.berkeley.edu/instructors.html

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Local search algorithmsIn many optimization problems, the path to the goal is irrelevant; the goal state itself is the solution

State space = set of "complete" configurations

Find optimal configuration, e.g., TSPFind configuration satisfying constraints, e.g., timetable, n-queens

In such cases, we can use local search algorithms (iterative improvement algorithms)

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

Constant space, suitable for online as well as offline search

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Example: n-queens

Put n queens on an n × n board with no two queens on the same row, column, or diagonalMove a queen to reduce number of conflicts

Almost always solves n-queens problems almost instantaneouslyfor very large n, e.g., n=1million

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Hill-climbing search: 8-queens problem

h = number of pairs of queens that are attacking each other, either directly or indirectly h = 17 for the above state

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Hill-climbing search: 8-queens problem

A local minimum with h = 1

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Example: Travelling Salesperson Problem

Start with any complete tour, perform pairwise exchanges

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

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Hill-climbing search

"Like climbing Everest in thick fog with amnesia"

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Hill-climbing search

Problem: depending on initial state, can get stuck in local maxima

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Hill-climbing with 8-queens

• Gets stuck 86% of the time

• Solves only 14% of problem instances

– 4 steps on average when it succeeds

– 3 steps on average when it gets stuck

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Hill climbing fixes

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Sideways Moves

escape from shoulders

loop on at maxima Limit number of consecutive sideways steps

100 consecutive sideways moves in 8-queensNow Solves 94% of problem instances (up from 14%)Average 21 moves when succeeds Average 64 moves when it fails

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Stochastic Hill Climbing

• Choose randomly among the uphill moves

• Or probability of selection based on steepness of move

• Converges more slowly, but can find better solutions

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First Choice Hill-climbing

• Implementation of stochastic hill-climbing

• Generating successors randomly until it finds one which is uphill– Very good when a state has many successors– Don’t waste time generating them all

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Random-restart hill climbing

series of hill-climbing searches (from randomly generated initial states)

overcomes local maxima trivially complete

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8-queens with Random Restartsp is probability of success

Number of restarts is 1/p8 queens, no sideways, p=0.14, so ~7 restarts8 queens, sideways, p=0.94, so ~1.06 restarts

Number of steps(1 x cost of success) + ((1-p)/p) x cost of failure)8 queens, no sideways, (1 x 4) + ((.86/.14) x 3) ~ 22 steps8 queens, sideways, (1x21) + ((0.06/0.94) x 64) ~ 25 steps

For 3 million queens, random restart finds solutions in under a minute

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Hill Climbing Summary

• The success of hill climbing depends very much on the shape of the state-space landscape.

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Simulated annealing search

Idea: escape local maxima by allowing some "bad" moves but gradually decrease their size and frequency

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Properties of simulated annealing search

One can prove: If T decreases slowly enough, then simulated

annealing search will find a global optimum with probability approaching 1

Intuition: Shake hard enough to dislodge from local minima but not hard enough to dislodge from global minima

Widely used in VLSI layout, airline scheduling, etc

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Local beam search

Keep track of k states rather than just one, choose top k of all their successors

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.

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Same as k random restarts? NO

In a local beam search, useful information is passed among the parallel search threads.

But can quickly become concentrated in a small region

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Stochastic Beam Search

Choose k successors randomly, biased towards good ones

Observe the close analogy to natural selection!

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Genetic algorithms 1= stochastic local beam search + generate successors from pairs of states

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Genetic algorithms 2A successor state is generated by combining two parent states

Start with k randomly generated states (population)

A state is represented as a string over a finite alphabet (often a string of 0s and 1s)

Evaluation function (fitness function). Higher values for better states.

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

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Genetic Algorithms 3

GAs frequently use states encoded as strings (GPs use programs - trees)

Crossover helps iff substrings are meaningful components