22c:145 artificial intelligence
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22c:145 Artificial IntelligenceFall 2005
Informed Search and Exploration III
Cesare Tinelli
The University of Iowa
Copyright 2001-05 — Cesare Tinelli and Hantao Zhang. a
a These notes are copyrighted material and may not be used in other course settings outside of the
University of Iowa in their current or modified form without the express written permission of the copyright
holders.22c:145 Artificial Intelligence, Fall’05 – p.1/17
Readings
Chap. 4 of [Russell and Norvig, 2003]
22c:145 Artificial Intelligence, Fall’05 – p.2/17
Iterative ImprovementAlgorithms
In many optimization problems the goal state itself is thesolution.
The state space is a set of complete configurations.
Search is about finding the optimal configuration (as in TSP) orjust a feasible configuration (as in scheduling problems).
In such cases, one can use iterative improvement (or localsearch) algorithms.
An evaluation (or objective) function h must be available thatmeasures the quality of each state.
Main Idea: Start with an initial configuration and make smallchanges to it that improve its quality.
22c:145 Artificial Intelligence, Fall’05 – p.3/17
Local Search: The LandscapeMetaphor
Ideally, the evaluation function h should be monotonic: thecloser a state to an optimal goal state the better its h-value.
Each state can be seen as a point on a surface.
The search consists in moving on the surface, looking for itshighest peaks (or, lowest valleys): the optimal solutions.
evaluation
currentstate
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Local Search Example: TSP
TSP: Travelling Salesperson Problem
h = length of the tour
Strategy: Start with any complete tour, perform pairwiseexchanges
22c:145 Artificial Intelligence, Fall’05 – p.5/17
Local Search Example: TSP
TSP: Travelling Salesperson Problem
h = length of the tour
Strategy: Start with any complete tour, perform pairwiseexchanges
Variants of this approach get within 1% of optimal very quickly withthousands of cities.
22c:145 Artificial Intelligence, Fall’05 – p.5/17
Local Search Example:n-queens
Put n queens on an n× n board with no two queens on thesame row, column, or diagonal
h = number of conflicts
Strategy: Move a queen to reduce number of conflicts
h = 5 h = 2 h = 0
22c:145 Artificial Intelligence, Fall’05 – p.6/17
Local Search Example:n-queens
Put n queens on an n× n board with no two queens on thesame row, column, or diagonal
h = number of conflicts
Strategy: Move a queen to reduce number of conflicts
h = 5 h = 2 h = 0
Almost always solves n-queens problems almost instantaneouslyfor very large n, e.g., n = 106.
22c:145 Artificial Intelligence, Fall’05 – p.6/17
Hill-Climbing (or GradientDescent) Search
“Like climbing Everest in thick fog with amnesia”
function HILL-CLIMBING( problem) returns a state that is a local maximuminputs : problem, a problemlocal variables : current, a node
neighbor, a node
current←MAKE-NODE(INITIAL-STATE[problem])loop do
neighbor←a highest-valued successor of currentif VALUE[neighbor] < VALUE[current] then return STATE[current]current← neighbor
end
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Hill-Climbing: Shortcomings
Depending on the initial state, it can get stuck on local maxima.
It may converge very slowly.
In continuous spaces, choosing the step-size is non-obvious.
currentstate
objective function
state space
global maximum
local maximum
"flat" local maximum
shoulder
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Hill Climbing: Improvements
Various possible alternatives:
Restart from a random point in the space.
Expand up to m > 1 generations of descendants beforechoosing best node.
Allow down-hill moves sometimes.
Keep n > 1 nodes in the fring at each step.
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Beyond Hill Climbing:Simulated Annealing Search
Allows occasional down-hill steps, to minimize the probability ofgetting stuck in a local maximum.
Down-hill steps taken randomly but with probability thatdecreases with time.
Probability controlled by a given annealing schedule for atemperature parameter T .
If schedule lowers T slowly enough, search will end in a globalmaximum.
It may take several tries with test problems to devise a goodannealing schedule.
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Simulated AnnealingAlgorithm
function SIMULATED-ANNEALING( problem, schedule) returns a solution stateinputs : problem, a problem
schedule, a mapping from time to “temperature”local variables : current, a node
next, a nodeT, a “temperature” controlling prob. of downward steps
current←MAKE-NODE(INITIAL-STATE[problem])for t← 1 to ∞ do
T← schedule[t]if T = 0 then return current
next← a randomly selected successor of current
∆E← VALUE[next] – VALUE[current]if ∆E > 0 then current← next
else current← next only with probability e∆ E/T
Erratum: In first if , replace “T = 0” with “T = 0 or IS-GOAL-STATE(current)”.22c:145 Artificial Intelligence, Fall’05 – p.11/17
Properties of SimulatedAnnealing
The smaller |∆E|, the greater e∆E
T ; the greater T , the greatere
∆E
T .
At fixed “temperature” T , state occupation probability reachesBoltzman distribution
p(x) = αeE(x)kT
T decreased slowly enough⇒ always reach best state.
This is not necessarily an interesting guarantee.
Devised by Metropolis et al., 1953, for physical processmodelling.
Widely used in VLSI layout, airline scheduling, etc.
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Beyond Hill Climbing: LocalBeam Search
Idea: keep k states instead of 1; choose top k of all theirsuccessors.
Not the same as k searches run in parallel!Searches that find good states recruit other searches to jointhem.
Problem: quite often, all k states end up on same localmaximum.
Solution: choose k successors randomly, biased towards goodones.
Observe the close analogy to natural selection.
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Beyond Hill Climbing: GeneticAlgorithms
Inspired by Darwin’s theory of natural selection.
Each state is seen as an individual in a population.
A genetic algorithm applies selection and reproductionoperators to an initial population.
The aim is to generate individuals that are most successful,according to a given fitness function.
It is basically a stochastic local beam search, but withsuccessors generated from pairs of states.
Local maxima are avoided by giving a nonzero chance ofreproduction to low-scoring individuals.
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The Basic Genetic Algorithm
function GENETIC-ALGORITHM( population, FITNESS-FN) returns an individualinputs: population, a set of individuals
FITNESS-FN, a function that measures the fitness of an individual
repeatparents SELECTION( population, FITNESS-FN)population REPRODUCTION( parents)
until some individual is fit enoughreturn the best individual in population, according to FITNESS-FN
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Genetic Algorithms: ClassicApproach
Each state is represented by a finite string over a finitealphabet.
Each character in the string is a gene.
The selection mechanism is randomized, with the probability ofselection proportional to the fitness measure.
Reproduction is accomplished by cross-over and mutation.
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(a)Initial Population
(b)Fitness Function
(c)Selection
(d)Cross−Over
(e)Mutation
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22c:145 Artificial Intelligence, Fall’05 – p.16/17
Encodings for GeneticAlgorithms
Choosing the right encoding of state configurations to strings iscrucial.
Crossover helps only if substrings are meaningful components thatcan be reassembled into a new meaningful configuration.
+ =
Note: GAs are not evolution: e.g., real genes encode replicationmachinery!
22c:145 Artificial Intelligence, Fall’05 – p.17/17
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