Computational complexity and Computational complexity and

simulation of rare events of simulation of rare events of IsingIsing spin glassesspin glasses

PelikanPelikan, M., , M., OcenasekOcenasek, J., , J., TrebstTrebst, S., Troyer, M., , S., Troyer, M., AletAlet, F. , F.

MotivationMotivation

Spin glassSpin glassOrigin in physics, but interesting for optimization as wellOrigin in physics, but interesting for optimization as well

Huge number of local optima and plateausHuge number of local optima and plateausLocal search fails miserablyLocal search fails miserablySome classes can be Some classes can be scalablyscalably solved using analytical methodssolved using analytical methodsSome classes provably NPSome classes provably NP--completecomplete

This paperThis paperExtends previous work to more classes of spin glassesExtends previous work to more classes of spin glassesProvides a thorough statistical analysis of resultsProvides a thorough statistical analysis of results

OutlineOutline

Hierarchical BOA (Hierarchical BOA (hBOAhBOA))Spin glassesSpin glasses

DefinitionDefinitionDifficultyDifficultyConsidered classes of spin glassesConsidered classes of spin glasses

ExperimentsExperimentsSummary and conclusionsSummary and conclusions

Hierarchical BOA (Hierarchical BOA (hBOAhBOA))

PelikanPelikan, Goldberg, and Cantu, Goldberg, and Cantu--Paz (2001, 2002)Paz (2001, 2002)Evolve population of candidate solutionsEvolve population of candidate solutionsOperatorsOperators

SelectionSelectionVariationVariation

Build a Bayesian network with local structures for selected soluBuild a Bayesian network with local structures for selected solutionstionsSample the built network to generate new solutionsSample the built network to generate new solutions

ReplacementReplacementRestricted tournament replacement for Restricted tournament replacement for nichingniching

hBOAhBOA: Basic algorithm: Basic algorithm

Current population Selection

New population

Bayesian network

Restricted tournament replacement

Spin glass (SG)Spin glass (SG)

Spins arranged on a lattice (1D, 2D, 3D)Spins arranged on a lattice (1D, 2D, 3D)Each spin Each spin ssii is +1 or is +1 or --11Neighbors connectedNeighbors connectedPeriodic boundary conditionsPeriodic boundary conditionsEach connection Each connection ((i,ji,j)) contains number contains number JJi,ji,j (coupling)(coupling)

Couplings usually initialized randomlyCouplings usually initialized randomly+/+/-- J couplings J couplings ~~ uniform on {uniform on {--1, +1}1, +1}Gaussian couplings Gaussian couplings ~~ N(0,1)N(0,1)

Finding ground states of Finding ground states of SGsSGs

EnergyEnergy

Ground stateGround stateConfiguration of spins that minimizes E for given couplingsConfiguration of spins that minimizes E for given couplingsConfigurations can be represented with binary vectorsConfigurations can be represented with binary vectors

Finding ground statesFinding ground statesFind ground states given couplingsFind ground states given couplings

∑><

=ji

jjii sJsE,

,

22--dimensional +/dimensional +/-- J SGJ SG

==

=

=

≠

≠

≠

≠≠≠

≠

≠

Spins:

≠ =

As constraint satisfaction problemAs constraint satisfaction problem

General caseGeneral casePeriodic boundary Periodic boundary condcond. (last and first connected). (last and first connected)Constraints can be weightedConstraints can be weighted

Constraints:

SG DifficultySG Difficulty

1D1DTrivial, deterministic Trivial, deterministic O(nO(n) algorithm) algorithm

2D2DLocal search fails miserably (exponential scaling)Local search fails miserably (exponential scaling)Good recombinationGood recombination--based based EAsEAs should scaleshould scale--upupAnalytical method exists, O(nAnalytical method exists, O(n3.53.5))

3D3DNPNP--completecompleteBut methods exist to solve But methods exist to solve SGsSGs of 1000s spinsof 1000s spins

Test SG classesTest SG classes

Dimensions n=6x6 to n=20x20Dimensions n=6x6 to n=20x201000 random instances for each n and distribution1000 random instances for each n and distribution2 basic coupling distributions2 basic coupling distributions

+/+/-- J, where couplings are randomly +1 or J, where couplings are randomly +1 or --11Gaussian, where couplings Gaussian, where couplings ~N(0,1)~N(0,1)

Transition between the distributions for n=10x10Transition between the distributions for n=10x104 steps between the bounding cases4 steps between the bounding cases

Coupling distributionCoupling distribution22--component normal mixture with overall component normal mixture with overall σσ22=1=1Vary Vary μμ22--μμ11 is from 0 to 2is from 0 to 2

-3 -2 -1 0 1 2 3

Pure Gaussian (μ=0)

-3 -2 -1 0 1 2 3

μ = 0.60

-3 -2 -1 0 1 2 3

μ = 0.80

-3 -2 -1 0 1 2 3

μ = 0.95

-3 -2 -1 0 1 2 3

μ = 0.99

-3 -2 -1 0 1 2 3

± J

( ) ( ) ( )2

,, 222

211 σμσμ NNJp +

=

Analysis of running timesAnalysis of running times

Traditional approachTraditional approachRun multiple times, estimate the meanRun multiple times, estimate the meanOften works well, but sometimes misleadingOften works well, but sometimes misleading

Performance on Performance on SGsSGsMCMC performance shown to follow MCMC performance shown to follow FrechetFrechet distr.distr.All distribution moments illAll distribution moments ill--defined (incl. the mean)!defined (incl. the mean)!

HereHereIdentify distribution of running timesIdentify distribution of running timesEstimate parameters of the distributionEstimate parameters of the distribution

FrechetFrechet distributiondistribution

Central limit theorem for Central limit theorem for extremalextremal valuesvalues

ξξ = shape, = shape, μμ = location, = location, ββ = scale= scaleξξ determines speed of tail decaydetermines speed of tail decayξξ<0: <0: FrechetFrechet distribution (polynomial decay)distribution (polynomial decay)ξξ=0: =0: GumbelGumbel distribution (exponential decay)distribution (exponential decay)ξξ>0: >0: WeibullWeibull distribution (faster than exponential decay)distribution (faster than exponential decay)

FrechetFrechet: : mmthth moment exists moment exists iffiff ||ξξ|<m |<m

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛ −+−=

ε

βμξ βμξ

1

;; 1exp xH

Our case

ResultsResults

+/+/-- J vs. Gaussian couplingsJ vs. Gaussian couplingsDistribution of the number of evaluationsDistribution of the number of evaluationsLocation scaleLocation scale--upupShapeShape

TransitionTransitionLocation changeLocation changeShape changeShape change

10 independent runs for each instance10 independent runs for each instanceMinimum population size to converge in all runsMinimum population size to converge in all runs

Number of evaluationsNumber of evaluations

Location, Location, μμ

Shape, Shape, ξξ

Transition: Location & ShapeTransition: Location & Shape

DiscussionDiscussion

Performance on +/Performance on +/-- J J SGsSGsNumber of evaluations grows approx. as O(nNumber of evaluations grows approx. as O(n1.51.5))Agrees with BOA theory for uniform scalingAgrees with BOA theory for uniform scaling

Performance on Gaussian Performance on Gaussian SGsSGsNumber of evaluations grows approx. as O(nNumber of evaluations grows approx. as O(n22))Agrees with BOA theory for exponential scalingAgrees with BOA theory for exponential scaling

TransitionTransitionTransition is smooth as expectedTransition is smooth as expected

Important implicationsImportant implications

Selection+RecombinationSelection+Recombination scales up greatscales up greatExponential number of optima easily escapedExponential number of optima easily escapedGlobal optimum found reliablyGlobal optimum found reliablyOverall time complexity similar to best analytical Overall time complexity similar to best analytical methodmethod

Selection+MutationSelection+Mutation fails to scale upfails to scale upEasily trapped in local minimaEasily trapped in local minimaExponential scalingExponential scaling

ConclusionsConclusions

Average running time anal. might be insufficientAverage running time anal. might be insufficientInIn--depth statistical analysis confirms past resultsdepth statistical analysis confirms past resultshBOAhBOA scales up well on all tested classes of scales up well on all tested classes of SGsSGshBOAhBOA scalability agrees with theoryscalability agrees with theoryPromising direction for solving other Promising direction for solving other challenging constraint satisfaction problemschallenging constraint satisfaction problems

ContactContact

Martin Martin PelikanPelikanDept. of Math and Computer Science, 320 CCBDept. of Math and Computer Science, 320 CCBUniversity of Missouri at St. LouisUniversity of Missouri at St. Louis8001 Natural Bridge Rd.8001 Natural Bridge Rd.St. Louis, MO 63121St. Louis, MO 63121

EE--mail: mail: pelikan@cs.umsl.edupelikan@cs.umsl.edu

WWW:WWW: http://http://www.cs.umsl.edu/~pelikanwww.cs.umsl.edu/~pelikan//