oftheagentstoacommonvalue.Inparticular,weprovethatallagentshavethesame - PDF document

oftheagentstoacommonvalue.Inparticular,weprovethatallagentshavethesamevalueaftertimesteps,whereisthenumberofquantizationlevelsperunitvalue.Duetotherounding-downfeatureofthequantizer,thisalgorithmdoesnotpreservetheaverageofthevaluesateachiteration.However,weprovideboundsontheerrorbetweentheÞnalconsensusvalueandtheinitialaverage,asafunctionofthenumberofavailablequantizationlevels.Inparticular,weshowthattheerrorgoesto0atarateof,asthenumberofquantizationlevelsincreasestoinÞnity.Otherthanthepaperscitedabove,ourworkisalsorelatedto[11]and[5],[6],whichstudiedtheeffectsofquantizationontheperformanceofaveragingalgorithms.In[11],Kashyapetal.proposedrandomizedgossip-typequantizedaveragingal-gorithmsundertheassumptionthateachagentvalueisanin-teger.Theyshowedthatthesealgorithmspreservetheaverageofthevaluesateachiterationandconvergetoapproximatecon-sensus.TheyalsoprovidedboundsontheconvergencetimeofthesealgorithmsforspeciÞcstatictopologies(fullyconnectedandlinearnetworks).Intherecentwork[5],Carlietal.proposedadistributedalgorithmthatusesquantizedvaluesandpreservestheaverageateachiteration.Theyshowedfavorableconver-gencepropertiesusingsimulationsonsomestatictopologies,andprovidedperformanceboundsforthelimitpointsofthegen- ,includingself-edges,suchthat.Ateachtime,thenodesÕconnectivitycanberepresentedbythedirectedgraph.Ourgoalistostudytheconvergenceoftheiteratestotheaverageoftheinitialvalues,,asproachesinÞnity.Inordertoestablishsuchconvergence,weimposesomeassumptionsontheweightsandthegraphsequence.Assumption1:Foreach,theweightmatrixisadoublystochasticmatrix1withpositivediagonalentries.Additionally,thereexistsaconstantsuchthatif,then analysis.Inparticular,inthenextsubsection,weexploresev-eralimplicationsofthedoublestochasticityassumptionontheweightmatrix.A.PreliminariesonDoublyStochasticMatricesWebeginbyanalyzinghowthesamplevariancechangeswhenthevectorismultipliedbyadoublystochasticmatrix.Thenextlemmashowsthat.Thus,underAssumption1,thesamplevarianceisnonin-creasingin,andcanbeusedasaLyapunovfunction.Lemma4:Letbeadoublystochasticmatrix.Then,forallwhereisthethentryofthematrix.Proof:Letdenotethevectorinwithallentriesequalto1.Thedoublestochasticityofimplies2Inthesequel,thenotationwillbeusedtodenotethedoublesum.Notethatmultiplicationbyadoublystochasticmatrixpre-servestheaverageoftheentriesofavector,i.e.,forany,thereholdsWenowwritethequadraticformexplicitly,asfollows:bethethentryof.Notethatissymmetricandstochastic,sothatand.Then,itcanbeveriÞedthat(3)whereisaunitvectorwiththethentryequalto1,andallotherentriesequalto0(seealso[22]whereasimilardecompo-sitionwasused).Bycombining(2)and(3),weobtainNotethattheentriesofarenonnegative,becausetheweightmatrixhasnonnegativeentries.Inview stepinasthesecondstepin.ratheronallassociatedwithedgesthatcrossapar-ticularcutinthegraph.Forsuchgroupsof,weprovealowerboundwhichislinearin,asseeninthefol-lowing.Lemma5:Letbearow-stochasticmatrixwithpositivediagonalentries,andassumethatthesmallestpositiveentryinisatleast.Also,letbeapartitionofthesetintotwodisjointsets.If weseethatthereexistsanedgeinthesetthatcrossesthecut.Letbesuchanedge.Withoutlossofgenerality,weassumethatand.WedeÞneSeeFig.1(a)foranillustration.Sinceisarow-stochasticma- e)Tosimplifynotation,let.Byassumption,wehave.Wemaketwoobservations,asfollows:1)Supposethat.Then,forsome,wehaveeitheror.Becauseisnonnegativewithpositivediagonalentries,wehaveandbyLemma5,weobtain(10)2)Fixsome,with Wenextestablishaboundonthevariancedecreasethatplaysakeyroleinourconvergenceanalysis.Lemma9:LetAssumptions1and6hold,andsupposethat.ThenProof:Withoutlossofgenerality,weassumethatthecom-ponentsofhavebeensortedinnonincreasingorder.ByLemma8,wehaveThisimpliesthatObservethattheright-handsidedoesnotchangewhenweadd Considerthebalancingalgorithm,andsupposethatisasequenceofundirectedgraphssuchthatisconnected,forallintegers.ThereexistsanabsoluteconstantsuchthatwehaveProof:Notethatwiththisalgorithm,thenewvalueatsomeisaconvexcombinationofthepreviousvaluesofitselfanditsneighbors.Furthermore,thealgorithmkeepsthesumofthenodesÕvaluesconstant,becauseeveryacceptedofferin-volvesanincreaseatthereceivingnodeequaltothedecreaseattheofferingnode.Thesetwopropertiesimplythatthealgorithmcanbewrittenintheformisadoublystochasticmatrix,determinedbyand.Itcanbeseenthatthediagonalentriesofarepositiveand,furthermore,allnonzeroentriesofarelargerthanorequalto1/3;thus,.Weclaimthatthealgorithm[inparticular,thesequence]satisÞesAssumption6.Indeed,supposethatattime,thenodesarereorderedsothatthevaluesarenonincreasingin.Fixsome,andsupposethat.Let smallest.Since,wehaveLet.Clearly,forall,and.Moreover,themonotonicityofimpliesthemonotonicityofThusNext,wesimplyrepeatthestepsofLemma9.Wecanassumewithoutlossofgeneralitythat.DeÞneforand.Wehavethatareallnonnegativeand.ThereforeTheminimizationproblemontheright-handsidehasanoptimalvalueofatleast,andthedesiredresultfollows.C.ExtensionsandModiÞcationsInthissubsection,wecommentbrießyonsomecorollariesofTheorem17.First,wenotethattheresultsofSectionIVimmediatelycarryovertothequantizedcase.Indeed,inSectionIV,weshowedhowtopicktheweightsinadecentralizedmanner,basedonlyonlocalinformation,sothatAssumptions1and6aresat-isÞed,with.Whenusingaquantizedversionofthebal-ancingalgorithm,weonceagainmanagetoremovethefactoroffromourupperbound.Proposition18:Forthequantizedversionofthebalancingalgorithm,andunderthesameassumptionsasinTheorem12,if,then thefollowingproperty.Foranynonnegativeinteger,,,andand,thereexistasequenceofweightmatricessatisfyingAssumptions1and2,andaninitialvaluesatisfyingAssumption13,andanumberquantizationlevels(dependingon)suchthatunderthedynamicsof(14),if,thenProof: thecommonlimitofthevaluesgeneratedbythequan-tizedalgorithm(14),wehaveProof:ByProposition19,afteriterations,allnodeswillhavethesamevalue.Sinceandtheaveragedecreasesbyatmostateachiteration,theresultfollows.Letusassumethattheparameters,,andareÞxed.Proposition21impliesthatasincreases,thenumberofbitsusedforeachcommunication,whichisproportionalto,needstogrowonlyastomaketheerrornegligible.Furthermore,thisistrueeveniftheparameters,,and 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