!(tP.TheeigenvaluesofParereal(sinceitissymmetric),andbyPerron - PDF document

Pij=0,( !(tP.TheeigenvaluesofParereal(sinceitissymmetric),andbyPerronÐFrobeniustheory,nomorethan1inmagnitude.Wewilldenotetheminnonincreasingorder:1="1(P)%"2(P)%ááá%"n(P)%&1.P:µ( =maxS#V!!!Prob!(S)&Prob÷!(S)!!!=12"i|#i&÷#i|(see,e.g.,[13,section4.1.1]).Wehavethefollowingboundonthetotalvariationdistancebetween 1andthedistributionconvergestouniformasymptoticallyasµt.Wecallthequantitylog(1/µ)themixingrate,and$=1/log(1/µ)themixingtime.Themixingtime$givesanasymptoticmeasureofthenumberofstepsrequiredforthetotalvariationdistanceofthedis-tributionfromuniformtobereducedbythefactore.IftheSLEMisverycloseto1,themixingratelog(1/µ)isapproximately1&µ,whichisoftenreferredtoasthespectralgapintheliterature.Themixingrate,mixingtime,andthespectralgapcanallserveasthemeasureforfastmixing.SincetheyareallmonotoneintheSLEM,wewillfocusontheSLEM dibethedegreeofvertexi Prwji)/ i,k)$Emax{0,1/di&1/dk rithmcanresultonlyifthenumberofsimulationstepsisreasonablysmall,whichusuallymeansdramaticallylessthanthesizeofthestatespaceitself.Forexample,aMarkovchainiscalledrapidlymixingifthesizeofstatespaceisexponentialinsomeinputdatasize,whereasthemixingtimeisboundedbyapolynomial.MostpreviousworkfocusedonboundingtheSLEM(orthespectralgap)ofaMarkovchainwithvarioustechniquesanddevelopingsomeheuristicstoassigntran-sitionprobabilitiestoobtainfastermixingMarkovchains.Somewell-knownanalyticapproachesforboundingtheSLEMare:couplingmethodsandstrongstationarytimes[1,16],conductance[29],geometricbounds[20],andmulticommodityßows[48,30].DiaconisandSalo!-Coste[19]surveyedwhatisrigorouslyknownaboutrunningtimesoftheMetropolisÐHastingsalgorithmÑthemostcelebratedalgorithmforconstruct-ingMarkovchainsinMonteCarlomethods.Kannan[31]surveyedtheuseofrapidlymixingMarkovchainsinrandomizedpolynomialtimealgorithmstoapproximatelysolvecertaincountingproblems.MorebackgroundonfastmixingMarkovchainscanbefoundin,e.g.,[13,4,44]andreferencestherein. 1.4.Outline.Insection2,weshowthattheFMMCproblemcanbecastasaconvexoptimizationproblem,andevenmorespeciÞcally,asasemideÞniteprogram "1(P1,wecanexpressthesecondlargesteigenvalueas"2(P)=sup{uTPu|'u' (P)=max{"2(P),&"n( ,and'á'2denotesthespectralnorm,ormaximumsingularvalue.(Inthiscase,sincethematricesaresymmetric,'á'2isthelargesteigenvaluemagnitude.)Theformula(4)givesµ(P)asthenormofana"nefunctionof P&(1/n)11T (i,j)/#E.(6)HerethevariablesarethematrixPandthescalars 2/2!2/2!2/2,-1 ,wherer#Znr+c#Znc+Xasthevertexset.Wesaythattwotables(matrices)Xand÷XX)areadjacent(connectedbyanedge)ifX&÷X=(ei&ej)( 2each.ThismodiÞcationdoesnÕtchangetherowandcolumnsums.IfthemodiÞcationresultsinnegativeentries,wediscardit(andstayatthecurrenttable);otherwise,weacceptthemodiÞcation(jumptoanadjacenttable).Wethenrepeattheprocess,byselectingnewpairsofrowsandcolumns.ThisdescribesaMarkovchainonX,anditcanbeshownthatthischain(graph)isconnected.Actu-ally,thisispreciselythemaximum-degreeMarkovchainonthegraphofcontingencytables,anditgeneratesuniformsamplingofthetablesinthesteadystate.See[17]forareviewofthebackgroundandapplicationsofcontingencytables.Whilethemaximum-degreechainseemstobetheonlypracticalmethodthatcanbeimplementedtocarryoutuniformsamplingonasetofcontingencytables,itisofacademicinteresttocompareitsmixingratewiththatoftheMetropolisÐ usethemaximum-degreeheuristictogetauniformdistribution.Wehopetoapplyourmethodstotheseproblemstogetfasteralgorithms.3.3.ARandomFamily.Wegenerateafamilyofgraphs,allwith50vertices,asfollows.FirstwegenerateasymmetricmatrixR#R50"50,whoseentriesRij,fori(j,areindependentanduniformlydistributedontheinterval[01].Foreachthresholdvaluec#[0,1]weconstructagraphbyplacinganedgebetweenverticesiandjfori*=jifRij(c.Wealwaysaddeveryself-looptothegraph.Byincreasingcfrom0to1,weobtainamonotonefamilyofgraphs;i.e.,thegraphassociatedwithalargervalueofccontainsalltheedgesofthegraphassociatedwithasmallervalueofc.Weonlyconsidervaluesofcabovethesmallestvaluethatgivesaconnectedgraph.Foreachgraph,wecomputetheSLEMsofthemaximum- Ð0.8Ð0.6Ð0.4Ð0.200.20.40.60.8102468maximum-degree chain .Thesumofthesingularvaluesofasymmetricmatrixisanorm;indeed,itisthedualnormofthespectralnorm,sowedenoteitby'á'&.ThedualFMMCproblemisconvex,sincetheobjective,whichismaximized,islinear,henceconcave,andtheconstraintsareallconvex. Y,zaredualfeasible,then1Tz(µ(P).(9)WeprovethisbyboundingTrY(P&(1/n)11T)fromaboveandbelow.BythedeÞnitionofdualnorm,wehaveTrY* TrY*P&(1/n)11T+=TrYP="i,jYijPij%"i,j(1/2)(zi+zj =1Tz.TheÞrstequalityusesthefactthatY1Pij=0for(i,j)/#Eand(1/2)(zi+zj)(Yij .11/,Y P#isoptimalifandonlyifthereexistdualvariablesz#andY# (i,j)/#E.¥Dualfeasibility.Y# G=vvTisasubgradientofµ(PvT1=0.BythevariationalcharacterizationofthesecondeigenvalueofPand÷P,wehaveµ(P)="2(P)= µ(P)v, ++TrY'=1 E(l E(m)u6,withcomponentsgl( p))andvisauniteigenvectorassociatedwiththeeigenvalue"n( pl:=p pontoonehalf-spaceatatime,andeachprojectionisveryeasytocompute.Duringeachexecutionoftheinnerloop(thewhileloop),(13)updatesI(iiwithstrictlypositivetransitionprobabilities,and|I(i)|isitscardinality.Ifpsum=)l$I(i)pl�1,wewouldliketoprojectpontothehalf-space)l$I(i)pl ontothehalf-space"l$I(i)pl(psum&'|I(i)|,(17)where'ischosentoavoidnegativecomponentsoftheprojection;see(14).Theprojectionstep(15)isverysimple.Theright-handsideof(17)isatleast1,anditiseasytoverifythatthewhileloopterminatesinaÞnitenumberofsteps,boundedbythedegreeofthenode.Moreover,everyhalf-spaceoftheform(17)containsthefeasibleset{p|p%0,Bp(1}.Thisimpliesthatthedistancefromanyfeasiblepointisreducedbyeachprojection.OncethesumprobabilityconstraintissatisÞedatanode,itwillneverbede-stroyedbylaterprojectionsbecausetheedgeprobabilitiescanonlydecreaseinthesequentialprojectionprocedure.Letpdenotetheprobabilityvectorafterstep1,andp+denotethevectorafterstep2.Itisclearthatp+producedbythesequentialprojectionmethodisalways Pandtherank-1propertyof11Tfore"&k=1/)kandstartthetransitionmatrixattheMetropolisÐHastingschain.TheprogressofthealgorithmisplottedinFigure4,whichshowsthemagnitudeofthetwoextremeeigenvalues"2and"n ProgressofthesubgradientmethodfortheFMMCproblemwithagraphwith #P=PT#,whichmeansthatthematrix#1/2 0.20.40.60.8 values±µforeachMarkovchain.ofcourse,hasthesameeigenvaluesasP).Theeigenvectorof#1/2P#'1/2associatedwiththemaximumeigenvalue1isq=()!1,...,)!n).TheSLEMµ(P#1/2P#'1/2or,equivalently,itsspectralnormrestrictedtothesubspaceq%.Thiscanbewrittenasµ(P)='(I 1/2P#'1/2&qqT'2.ThusthefastestmixingreversibleMarkovchainproblemcanbeformulatedasminimizeµ(P)='#1/2P#'1 ,asin(6).7.Extensions.7.1.ExploitingSymmetry.Inmanycases,thegraphsofinteresthavelargesymmetrygroups,andthiscanbeexploitedtosubstantiallyincreasethee"ciencyofsolutionmethodsoreven,insomecases,tosolvetheFMMCproblemanalytically.Thisisexploredinfarmoredetailin[10,42];herewedescribeaverysimplecasetoillustratethebasicidea.WeÞrstobservethatifagraphissymmetric,thenwecanassumewithoutlossofgeneralitythattheoptimaltransitionmatrixP#isalsosymmetric.Toseethis,letP# underthesymmetrygroup. dmax=n.Analysispresentedin[10]showsthat1&µ(Pmd)=2n'2+O( (EP,"(f,f)Var"(f)!!!!!Var"( !-diagonalentriesofP.Inparticular,sincePmhij%Pmdiji*=j,wealwayshave"2(Pmh)("2(Pmd),"n(Pmh)( foranysymmetrictransitionprobabilitymatrixPdeÞnedonthegraph.Thus,thespectralgapoftheoptimalchainisnomorethanafactordmax+1largerthanthespectralgapofthemodiÞedmax-degreechain.7.3.Optimizinglog-SobolevConstants.Wehaveusedthespectralgapasa &=inf(EP,"(f,f)L"(f)!!!!!L"(f) "(f)="i|fi|2log9 ,IntroductiontoMarkovChains,withSpecialEmphasisonRapidMixing,Adv.LecturesMath.,Vieweg,Weisbaden,Germany,2000.[5]A.Ben-TalandA.Nemirovski,LecturesonModernConvexOptimization:Analysis,Al-gorithms,andEngineeringApplications,MPS/SIAMSer.Optim.2,SIAM,Philadelphia,2001.[6], P.Br ,Math.ProgrammingStud.,3(1975),pp.35Ð55.[16]P.Diaconis,GroupRepresentationsinProbabilityandStatistics,InstituteofMathematicalStatistics,Hayward,CA,1988.[17]P.DiaconisandA.Gangolli Eigenvalueoptimization,ActaNumer.,5(1996),pp.149Ð190.[35]J.Liu [48]A.Sinclair,ImprovedboundsformixingratesofMarkovchainsandmulticommodityßow,Combin.Probab.Comput.,1(1992),pp.351Ð370.[49]A.Sinclair,AlgorithmsforRandomGenerationandCounting:AMarkovChainApproach,Birkh¬auserBoston,Boston,1993.[50]D.Stroock,LogarithmicSobolevinequalitiesforGibbsstates,inDirichletForms(Varenna,1992),LectureNotesinMath.1563,Springer-Verlag,Berlin,1993,pp.194Ð228.[51]J.F.Sturm,UsingSeDuMi1.02