LiftedMAPInferenceforMarkovLogicNetworks - PDF document

LiftedMAPInferenceforMarkovLogicNetworks allfalse.ThishelpsustofurtherreducethecomplexityofinferencefromO(Qni=1(di+1))toO(2n),i.e.,thecom-plexityofinferenceisindependentofthedomainsize.Weutilizetheaforementionedresultsbydevelopingamap-pingfromtheliftedsearchspacetopropositionalsearchspaceofthesamesize.ThishelpsusreformulatetheMAPinferencetaskovernon-sharedMLNsasproposi-tionalMAPinferenceoveraMarkovnetworksuchthat:(i)thenumberofrandomvariablesintheMarkovnetworkisequaltothenumberofatomsintheMLN(i.e.,equalton)and(ii)thedomainsizeofeachrandomvariableisei-therdior2dependinguponwhetherthenon-sharedMLNhasself-joinsornot.Thetwokeyfeaturesofthisformula-tionare:(i)wecanpluginanyknownpropositionalMAPinferencealgorithmforinferenceonthisMarkovnetwork;and(ii)sincethepropositionalalgorithmoperatesintheliftedsearchspace,ithasthesameperformanceguaranteesasaliftedalgorithm.Thus,bypluggingindifferentpropo-sitionalMAPinferencealgorithms,ourapproachyieldsafamilyofliftedMAPinferencealgorithms.Ourapproachisquitegeneralandcanbeextendedtoar-bitraryMLNsthathavesharedtermsinastraight-forwardway:simplygroundallthesharedtermsoftheMLNtoobtainanequivalentnon-sharedMLN.TheonlycaveatisthatifalltermsinallatomsoftheMLNareshared,ourapproachwillhavethesamecomplexityasgroundinferencebutneverworse(e.g.,inthetransitiveformula8x;y;zR(x;y)^R(y;z))R(x;z),alltermsofRareshared).Notethatthisisalimitationofnotonlyourap-proachbutliftedinferenceingeneral:itisonlyusefulwhensymmetriesarepresent.Weexperimentallyevaluatedourapproachonthreebench-markMLNs:WebKBandInformationExtractionMLNsavailablefromtheAlchemy[12]webpageandtheStu-dentMLNcreatedbyus.Weusedtwostate-of-the-artMAPinferencealgorithmswithinourapproach:(i)Gurobi[8],whichisanintegerlinearprogrammingsolverand(ii)MaxWalkSAT[11]whichisapopularlocalsearchsolver.Ourexperimentsclearlyshowthatourapproachissignif-icantlybetterintermsofsolutionqualityandscalabilitythangroundinference.Inparticular,asweincreasethenumberofobjectsintheMLN,ouralgorithmsareanorderofmagnitudefasterandbetterintermsofsolutionquality.Therestofthepaperisorganizedasfollows.Insection2,wepresentpreliminaries.Insection3,wepresentournewapproachandinsection4,weextenditwithseveralheuristicsandpruningtechniques.Insection5,wepresentexperimentalresultsandconcludeinsection6.2NotationandPreliminariesInthissection,wedescribenotationandpreliminariesonpropositionallogic,rst-orderlogic,MarkovlogicandMAPinference.Formoredetails,referto[5,9,13].2.1PropositionalandFirst-orderLogicThelanguageofpropositionallogicconsistsofatomicsen-tencescalledpropositionsoratoms,andlogicalconnec-tivessuchas^(conjunction),_(disjunction),:(negation),)(implication)and,(equivalence).EachpropositiontakesvaluesfromthebinarydomainfFalse,Trueg(orf0,1g).Apropositionalformulafisanatom,oranycom-plexformulathatcanbeconstructedfromatomsusinglogi-calconnectives.Forexample,A,BandCarepropositionalatomsandf=A_:B^Cisapropositionalformula.Aknowledgebase(KB)isasetofformulas.AworldisatruthassignmenttoallatomsintheKB.First-orderlogic(FOL)generalizespropositionallogicbyallowingatomstohaveinternalstructure;anatominFOLisapredicatethatrepresentsrelationsbetweenobjects.Apredicateconsistsofapredicatesymbol,denotedbyMonospacefonts(e.g.,Friends,Smokes,etc.),fol-lowedbyaparenthesizedlistofargumentscalledterms.Atermisalogicalvariable,denotedbylowercaseletters(e.g.,x,y,etc.),oraconstant,denotedbyuppercaseletters(e.g.,X,Y,etc.).Weassumethateachlogicalvariable,sayx,istypedandtakesvaluesoveraniteset(calleddomain)
x.ThelanguageofFOLalsoincludestwoquantiers:8(universal)and9(existential)whichexpresspropertiesofanentirecollectionofobjects.Aformulainrstorderlogicisapredicate(atom),oranycomplexsentencethatcanbeconstructedfromatomsusinglogicalconnectivesandquantiers.Forexample,theformula8xSmokes(x))Asthma(x)statesthatallpersonswhosmokehaveasthma.Whereas9xCancer(x)statesthatthereexistsapersonxwhohascancer.Inthispaper,weuseasubsetofFOLwhichhasnofunc-tionsymbols,equalityconstraintsorexistentialquanti-ers.Wealsoassumethatdomainsarenite(andthere-forefunction-free)andthatthereisaone-to-onemappingbetweenconstantsandobjectsinthedomain(Herbrandin-terpretations).Weassumethateachformulafisoftheform8xf,wherexarethesetofvariablesinfandfisaconjunctionordisjunctionofliterals;eachliteralbe-inganatomoritsnegation.Forbrevity,wewilldrop8fromalltheformulas.Givenvariablesx=fx1;:::;xngandconstantsX=fX1;:::;XngwhereXi2
xi,f[X=x]isobtainedbysubstitutingeveryoccurrenceofvariablexiinfwithXi.Agroundformulaisaformulaob-tainedbysubstitutingallofitsvariableswithaconstant.AgroundKBisaKBcontainingallpossiblegroundingsofallofitsformulas.Forexample,thegroundingofaKBcontainingoneformula,Smokes(x))Asthma(x)where
x=fAna;Bobg,isaKBcontainingtwoformu-las:Smokes(Ana))Asthma(Ana)andSmokes(Bob))Asthma(Bob). SomdebSarkhel,DeepakVenugopal,ParagSingla,VibhavGogate Figure2:ThetotalweightofsatisedclausesasafunctionofcountingassignmentofRandS.Theplaneisforillustrationpur-poseonly.ofx;yisfA;B;C;D;Eg.Ifweiteratethroughallpos-siblecountingassignmentstoRandSandplotthetotalweightofsatisedclausesasafunctionofcountingassign-mentofRandSwegettheplotinFigure2.Figure2showsthatthefunctionintheplothasonlyfourextremepoints:(0;0);(0;5);(5;0)and(5;5).Theseextremepointscorre-spondtoallgroundingsofRandSaseitherbeingalltrueorallfalse.SincetheMAPvaluecanonlylieontheseex-tremepoints,weonlyhavetoevaluatetheseextremepointsforcomputingtheMAPvalue.ItturnsoutthattheMAPtupleish(R;0);(S;0)i.WeobserveinthepreviousexamplethatallthegroundatomsofthepredicateR(andthepredicateS)havethesametruthvalue.Wewillrefertothiskindofassignment(i.e.,allgroundatomshavingthesametruthvalue)asauniformassignment[1].Thisobservation,thattheatomshaveauniformassignmentintheMAPstate,holdsnotonlyforthisexamplebutforanynon-sharedMLNwithoutself-joins,andwewillprovethisformallynext.Lemma1.Thesumofweightsofsatisedclausesforanon-sharedMLNwithoutself-joinisamultilinearfunctiononthecountingassignmentofitspredicates.Proof.(Sketch)Consideranon-sharedMLNMthatcon-tainsmweightedclausesf(Ci;wi)gmi=1.LetV(Ci)repre-sentthesetofalltheatomsintheclauseCi.LetV(+)(Ci)representthesetofatomswhichappearaspositiveliteralsinCi.LetV(�)(Ci)representthesetofatomsappearingasnegativeliterals.GivenanatomR,let(R;vR)denoteitscountingassignment.Itcanbeeasilyshownthatthenum-berofgroundingsofCithatareunsatisedbythecountingassignmentisgivenby,YR2V(+)(Ci)(
R�vR)YR2V(�)(Ci)vRwhere
RrepresentsthenumberofpossiblegroundingsofR.Clearly,thetotalnumberofpossiblegroundingsofCiisequaltoQR2Ci(
R).Therefore,thesumofweightsofsatisedclausesforMisgivenby,XCiwi(YR2Ci(
R)�YR2V(+)(Ci)(
R�vR)YR2V(�)(Ci)vR)(4)Clearlyeq.4isamultilinearfunctioninvRsincevRneverappearsmorethanonceintheproductterm(iftherearenoself-joinsinM). Lemma2.Consideramultilinearfunction(v)denedoveratupleofvariablesv=(v1;v2;


;vn).Leteachvjtakevaluesfromthesetf0;1;2;



vjg.Then,atleastoneofthesolutionsvtotheoptimizationproblemargmaxv(v)issuchthateachvjliesattheextremesi.e.vj=0orvj=
j8j.Proof.Wewillprovethetheoremusinginductionovern,thenumberofvariablesoverwhichthemultilinearfunc-tionisdened.Clearly,thetheoremholdstrueforn=1sincealinearfunctionofonevariablehasitsmaximaattheextremes.Assumethetheoremholdsforanymultilin-earfunctiondenedovern�1variables.Considerthefunction(v)overthevariablesv=(v1;v2;


vn).Byre-arrangingterms,wecanwrite:maxv(v)=maxvnvn(maxvn(v))Since(v)isamultilinearfunction,itcanbeseenasalinearfunctionofvn(holdingothervariablesasconstant).Hence,theinnerexpressionontherightsideisoptimizedatanextremevalueofvn(vn=0orvn=
vn).Let0(vnvn)and
vn(vnvn),respectively,bethetwopossibleresultingfunctionsbysubstitutingthevaluesof0and
vn,forvnin(v).Inboththecases,wegetanewfunctionwhichismultilinearovern�1variables.Usingtheinductionhypothesis,itsmaximawilllieattheextremevaluesofvi;v2;


vn�1.Hence,oneofthemaximaoftheoriginalfunction(v)willlieattheextremevaluesofv1;v2;


vn.Hence,proved. Notethatabovelemmastatesthatatleastoneofthemax-imaofamultilinearfunctionwilllieatitsextremes.Itisstillpossiblethatthereareothermaximawhichdonotlieattheextremes(forinstance,thinkofaconstantfunction).Aslongasweareinterestedinndingoneofthem,abovelemmacanbeputtouse.Lemma1andLemma2allowustoproveoursecondmainresultstatedbelow.Theorem3.Foranon-sharedMLNwithoutself-joins,inatleastoneoftheMAPsolutions,allpredicateshaveuni-formassignments. SomdebSarkhel,DeepakVenugopal,ParagSingla,VibhavGogate (ii)WebKBMLN[12]fromtheAlchemywebpage,con-sistingofthreepredicatesandsixformulas.(iii)CitationInformation-Extraction(IE)MLN[12]fromtheAlchemywebpage,consistingofvepred-icatesandfourteenformulas.Inordertocomparetheperformanceandscalabilityofouralgorithms,werantwosetsofexperimentsillustratedinFig.3andFig.4.Fig.3,plotsthesolutionquality(to-talweightoffalseclauses)achievedbyeachalgorithmforvaryingtime-bounds.Fig.4plotstherelative-gapbetweentheoptimalsolutionandthesolutionoutputbyeachalgo-rithm,forvaryingdomain-sizes.Wedescribetheresultsofbothourexperimentsbelow.Allourexperimentswererunonaquad-coreCentOSmachinewith8GBRAM.5.1CostvsTimeTheresultsfortheStudentMLNareshowninFig.3(a)-(c).WeseethattheliftedalgorithmsL-ILPandL-MWSarethebestperformingalgorithmsforalldomain-sizes.Athigherdomain-sizes(100and500),thepropositionalsolversILP,MWSandTUFFYranoutofmemory.TheperformanceofL-MWSwassimilartoL-ILPfordomain-sizeequalto30.Fordomain-sizesof100and500,L-MWSgraduallyconvergestowardstheoptimalsolution,whereasL-ILPwasabletoexactlysolvetheprobleminlessthan10seconds.TheresultsforWebKBareshowninFig.3(d)-(f).Again,wecanseethattheliftedalgorithmsL-ILPandL-MWSoutperformthepropositionalalgorithmsandaremuchmorescalable.Forthelargerdomain-sizes(100and500),MWSandTUFFYrunoutofmemory.Fordomain-size30and100,theperformanceofboththeliftedalgorithmsL-ILPandL-MWSisquitesimilar.TheresultsforIEareshowninFig.3(g)-(i).Fordomain-size30,theperformanceofboththeliftedalgorithmsL-ILPandL-MWSisquitesimilar.Forthedomain-sizes100and500,L-ILPwasabletondtheoptimalsolutionwhileL-MWSwasfarfromoptimal.5.2AccuracyvsDomain-SizeFig.4illustratesthevariationinaccuracyforeachalgo-rithmasthedomain-sizeincreases.Here,wegaveeachalgorithmaxedtime-boundof500secondsandmeasuredtherelative-gapbetweentheoptimalsolution(opt)andthebestcostgivenbythealgorithm(c)usingjopt�cj opt.WeseethatbothL-MWSandL-ILParequiteaccurateandscaletomuchlargerdomain-sizes.Ontheotherhand,thereisanoticeabledropintheaccuracyofthepropositionalalgo-rithms,MWS,TUFFYandILPasweincreasethedomain-size.Forlargerdomain-sizes,thepropositionalalgorithmsrunoutofmemory.Insummary,ourexperimentsshowthatourtwoliftedal-gorithmsL-MWSandL-ILParefarmorescalableandac-curatethanpropositionalapproaches.Sincethetwoap-proachesarefundamentallydifferent,L-ILPisacompleteanytimesolverwhileL-MWSisanapproximatesolver,asexpectedtheyperformdifferentlyonthebenchmarks,withL-ILPbeingthesuperiorapproach.However,themainvirtueofourapproachisthatwecoulduseanyoff-the-shelfsolverthatispurelypropositionalinnaturetoperformliftedinference.Thisallowsustoscaletolargedomain-sizeswithoutimplementinganewliftedsolver.Webelievethatthisabstractiongreatlysimpliesthedevelopmentofliftedalgorithmsbybenettingfromtheadvancesmadeinpropositionalalgorithms.6SummaryandFutureworkInthispaper,weproposedageneralapproachforliftingMAPinferenceinMarkovLogicNetworks(MLNs).WeidentiedcasesinwhichwecanreduceliftedMAPinfer-encetoinferenceoveranequivalentpropositionaltheorysuchthatthenumberofpropositionalvariablesisequaltothenumberofrstorderatomsintheMLN.Weusedthisobservationinastraight-forwardmanner:converttheMLNtoanequivalentpropositionaltheoryandthenapplyanypropositionalalgorithmtosolveit.Forourexperiments,weusedtwopropositionalalgorithms,acomplete,any-timealgorithm(Gurobi)basedonIntegerLinearProgram-ming(ILP)andalocal-searchalgorithmcalledMaxWalk-sat.Ourexperimentsclearlydemonstratethescalabilityandpromiseofourapproach.Directionsforfutureworkinclude:combiningourap-proachwithotherliftedinferencerulessuchasthepowerrule[7,10];identifyingcaseswhereourgeneralizedthe-oremcanbeapplied;applyingtheresultsinthispapertoliftedMCMCapproaches[19];andusingourapproachforexploitingsymmetriesinprobabilisticgraphicalmodels.AcknowledgementsThisresearchwaspartlyfundedbyAROMURIgrantW911NF-08-1-0242,bytheAFRLundercontractnum-berFA8750-14-C-0021andbytheDARPAProbabilisticProgrammingforAdvancedMachineLearningProgramunderAFRLprimecontractnumberFA8750-14-C-0005.Theviewsandconclusionscontainedinthisdocumentarethoseoftheauthorsandshouldnotbeinterpretedasrepre-sentingtheofcialpolicies,eitherexpressedorimplied,ofDARPA,AFRL,AROortheUSgovernment.References[1]Apsel,U.;andBrafman,R.2012.ExploitingUniformAssignmentsinFirst-OrderMPE.InProceedingsofthe