计算语言学协会会刊, 2 (2014) 393–404. 动作编辑器: Robert C. 摩尔.
Submitted 2/2014; 修改 6/2014; 已发表 10/2014. C
(西德:13)
2014 计算语言学协会.
LocallyNon-LinearLearningforStatisticalMachineTranslationviaDiscretizationandStructuredRegularizationJonathanH.Clark∗ChrisDyer†AlonLavie†*MicrosoftResearch†CarnegieMellonUniversityRedmond,WA98052,USAPittsburgh,PA15213,USAjonathan.clark@microsoft.com{cdyer,alavie}@cs.cmu.eduAbstractLinearmodels,whichsupportefficientlearn-ingandinference,aretheworkhorsesofstatis-ticalmachinetranslation;然而,linearde-cisionrulesarelessattractivefromamodelingperspective.Inthiswork,weintroduceatech-niqueforlearningarbitrary,rule-local,non-linearfeaturetransformsthatimprovemodelexpressivity,butdonotsacrificetheefficientinferenceandlearningassociatedwithlinearmodels.Todemonstratethevalueofourtech-nique,wediscardthecustomarylogtransformoflexicalprobabilitiesanddropthephrasaltranslationprobabilityinfavorofrawcounts.Weobservethatouralgorithmlearnsavari-ationofalogtransformthatleadstobettertranslationqualitycomparedtotheexplicitlogtransform.Weconcludethatnon-linearre-sponsesplayanimportantroleinSMT,anob-servationthatwehopewillinformtheeffortsoffeatureengineers.1IntroductionLinearmodelsusinglog-transformedprobabilitiesasfeatureshaveemergedasthedominantmodelinMTsystems.ThispracticecanbetracedbacktotheIBMnoisychannelmodels(Brownetal.,1993),whichdecomposedecodingintotheproductofatranslationmodel(TM值)andalanguagemodel(LM),motivatedbyBayes’Rule.WhenOchandNey(2002)introducedalog-linearmodelfortrans-lation(alinearsumoflog-spacefeatures),theynotedthatthenoisychannelmodelwasaspecialcaseoftheirmodelusinglogprobabilities.This∗Thisworkwasconductedaspartofthefirstauthor’sPh.D.workatCarnegieMellonUniversity.sameformulationpersistedevenaftertheintroduc-tionofMERT(和,2003),whichoptimizesalin-earmodel;再次,usingtwologprobabilityfea-tures(TMandLM)withequalweightrecoveredthenoisychannelmodel.Yetsystemsnowusemanymorefeatures,someofwhicharenotevenprobabil-ities.Wenolongerbelievethatequalweightsbe-tweentheTMandLMprovidesoptimaltranslationquality;theprobabilitiesintheTMdonotobeythechainrulenorBayes’rule,nullifyingseveralthe-oreticalmathematicaljustificationsformultiplyingprobabilities.Thestoryofmultiplyingprobabilitiesmayjustamounttoheavilypenalizingsmallvalues.Thecommunityhasabandonedtheoriginalmo-tivationsforalinearinterpolationoftwolog-transformedfeatures.Isthereempiricalevidencethatweshouldcontinueusingthisparticulartrans-formation?Dowehaveanyreasontobelieveitisbetterthanothernon-lineartransformations?Toan-swerthese,weexploretheissueofnon-linearityinmodelsforMT.Intheprocess,wewilldiscusstheimpactoflinearityonfeatureengineeringandde-velopageneralmechanismforlearningaclassofnon-lineartransformationsofreal-valuedfeatures.Applyinganon-lineartransformationsuchaslogtofeaturesisonewayofachievinganon-linearresponsefunction,evenifthosefeaturesareaggre-gatedinalinearmodel.Alternatively,wecouldachieveanon-linearresponseusinganativelynon-linearmodelsuchasaSVM(Wangetal.,2007)orRankBoost(Sokolovetal.,2012).然而,MTisastructuredpredictionproblem,inwhichafullhypothesisiscomposedofpartialhypotheses.MTdecoderstakeadvantageofthefactthatthemodel
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scoredecomposesasalinearsumoverbothlocalfeaturesandpartialhypothesestoefficientlyper-forminferenceinthesestructuredspaces(§2)–cur-rently,therearenoscalablesolutionstointegratingthehypothesis-levelnon-linearfeaturetransformstypicallyassociatedwithkernelmethodswhilestillmaintainingpolynomialtimesearch.Anotheralter-nativeisincorporatingarecurrentneuralnetwork(Schwenk,2012;Aulietal.,2013;KalchbrennerandBlunsom,2013)oranadditiveneuralnetwork(Liuetal.,2013a).Whilethesemodelshaveshownpromiseasmethodsofaugmentingexistingmod-els,theyhavenotyetofferedapathforreplacingortransformingexistingreal-valuedfeatures.Inthisarticle,wediscussbackground(§2),de-scribelocaldiscretization,ourapproachtolearningnon-lineartransformationsofindividualfeatures,compareitwithgloballynon-linearmodels(§3),presentourexperimentalsetup(§5),empiricallyver-ifytheimportanceofnon-linearfeaturetransforma-tionsinMTanddemonstratethatdiscretizationcanbeusedtorecovernon-lineartransformations(§6),discussrelatedwork(§7),andconclude(§8).2BackgroundandDefinitions2.1FeatureLocality&StructuredHypothesesDecodingagivensourcesentencefcanbeex-pressedassearchovertargethypothesese,eachwithanassociatedcompletederivationD.Tofindthebest-scoringhypothesisˆe(F),alinearmodelappliesasetofweightswtoacompletehypothesis’featurevectorH:ˆe(F)=argmaxe,D|H|Xi=0wiHi(F,e,D)(1)然而,thishidesmanyoftherealitiesofperform-inginferenceinmoderndecoders.Traditionalin-ferencewouldbeintractableifeveryfeaturewereallowedaccesstotheentirederivationDanditsas-sociatedtargethypothesise.Decoderstakeadvan-tageofthefactthatfeaturesdecomposeoverpar-tialderivationsd.ForacompletederivationD,theglobalfeaturesH(D)areanefficientsummationoverlocalfeaturesh(d):ˆe(F)=argmaxe,D|H|Xi=0wiXd∈Dhi(d)|{z}Hi(D)(2)Thiscontrastswithnon-localfeaturessuchasthelanguagemodel(LM),whichcannotbeexactlycal-culatedgivenanarbitrarypartialhypothesis,whichmaylackbothleftandrightcontext.1Suchfeaturesrequirespecialhandlingincludingfuturecostesti-mation.Inthisstudy,welimitourselvestolocalfeatures,leavingthetraditionalnon-localLMfea-tureunchanged.Ingeneral,featurelocalityisrel-ativetoaparticularstructuredhypothesisspace,andisunrelatedtothestructuredfeaturesdescribedinSection4.2.2.2FeatureNon-LinearityandSeparabilityUnlikemodelsthatrelyprimarilyonalargenumberofsparseindicatorfeatures,state-of-the-artmachinetranslationsystemsrelyheavilyonasmallnumberofdensereal-valuedfeatures.However,unlikeindi-catorfeatures,real-valuedfeaturesmaybenefitfromnon-lineartransformationstoallowalinearmodeltobetterfitthedata.Decodersusealinearmodeltorankhypotheses,selectingthehighest-rankedderivation.Sincetheabsolutescoreofthemodelisirrelevant,non-linearresponsesareusefulonlyincaseswheretheyelicitnovelrankings.Inthissection,wewilldiscussthesecasesintermsofseparability.Here,wearesepa-ratingthecorrectlyrankedpairsofhypothesesfromtheincorrectintheimplicitpairwiserankingsde-finedbythetotalorderingonhypothesesprovidedbyourmodel.Whenthelocalfeaturevectorshofeachoracle-best2hypothesis(orhypotheses)aredistinctfromthoseofallothercompetinghypotheses,wesaythattheinputsareoracleseparablegiventhefeatureset.Ifthereexistsaweightvectorthatdistinguishestheoracle-bestrankingfromallotherrankingsunderalinearmodel,wesaythattheinputsarelinearlysep-arablegiventhefeatureset.Iftheinputsareora-cleseparablebutnotlinearlyseparable,wesaythattherearenon-linearitiesthatareunexplainedbythefeatureset.Forexample,thiscanhappenifafeatureispositivelyrelatedtoqualityinsomeregionsbutnegativelyrelatedinotherregions.Asweaddmoresentencestoourcorpus,sepa-rabilitybecomesincreasinglydifficult.Foragiven1Thisisespeciallyproblematicforchart-baseddecoders.2Wedefinetheoracle-besthypothesisintermsofsomeex-ternalqualitymeasuresuchasBLEU
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语料库,ifallhypothesesareoracleseparable,wecanalwaysproducetheoracletranslation–assum-inganoptimal(andpotentiallyverycomplex)modelandweightvector.Ifourhypothesisspacealsocon-tainsallreferencetranslations,wecanalwaysre-coverthereference.Inpractice,bothofthesecondi-tionsaretypicallyviolatedtoacertaindegree.How-ever,ifwemodifyourfeaturesetsuchthatsomelower-rankedhigher-qualityhypothesiscanbesepa-ratedfromallhigher-rankedlower-qualityhypothe-ses,thenwecanimprovetranslationquality.Forthisreason,webelievethatseparabilityremainsaninformativetoolforthinkingaboutmodelinginMT.Currently,non-linearitiesinnovelreal-valuedfea-turesaretypicallyaddressedviamanualfeatureen-gineeringinvolvingagooddealoftrialanderror(GimpelandSmith,2009)3orbymanuallydiscretiz-ingfeatures(e.g.indicatorfeaturesforcount=N).Wewillexploreonetechniqueforautomaticallyavoidingnon-linearitiesinSection3.2.3LearningwithLargeFeatureSetsWhileMERThasproventobeastrongbaseline,itdoesnotscaletolargerfeaturesetsintermsofbothinefficiencyandoverfitting.WhileMIRA(Chiangetal.,2008),Rampion(GimpelandSmith,2012),andHOLS(Flaniganetal.,2013)havebeenshowntobeeffectiveoverlargerfeaturesets,theyaredif-ficulttoexplicitlyregularize–thiswillbecomeim-portantinSection4.2.Therefore,weusethePROoptimizer(HopkinsandMay,2011)asourbaselinelearnersinceithasbeenshowntoperformcompa-rablytoMERTforasmallnumberoffeatures,andtosignificantlyoutperformMERTforalargenum-beroffeatures(HopkinsandMay,2011;Ganitke-vitchetal.,2012).OtherveryrecentMToptimiz-erssuchasthelinearstructuredSVM(CherryandFoster,2012),AROW(蒋,2012)andregular-izedMERT(Galleyetal.,2013)arealsocompatiblewiththediscretizationandstructuredregularizationtechniquesdescribedinthisarticle.43Gimpeletal.eventuallyusedrawprobabilitiesintheirmodelratherthanlog-probabilities.4Sincewedispensewithnearlyalloftheoriginaldensefea-turesandourstructuredregularizerisscalesensitive,onewouldneedtousethe‘1-renormalizedvariantofregularizedMERT.3DiscretizationandFeatureInductionInthissection,weproposeafeatureinductiontech-niquebasedondiscretizationthatproducesafeaturesetthatislesspronetonon-linearities(see§2.2).WedefinefeatureinductionasafunctionΦ(y)thattakestheresultofthefeaturefunctiony=h(X)∈Randreturnsatuplehy0,jiwherey0∈Risatransformedfeaturevalueandjisthetransformedfeatureindex.5Buildingonequation2,wecanapplyfeatureinductionasfollows:ˆe(F)=argmaxe,DXd∈D|H|Xi=0hy0,ji=Φi(你好(d))w0jy0|{z}H0(F,e,D)(3)Atfirstglance,onemightbetemptedtosim-plychoosesomenon-linearfunctionforΦ(e.g.log(X),经验值(X),罪(X),xn).然而,evenifweweretorestrictourselvestosome“standard”setofnon-linearfunctions,manyofthesefunctionshavehyperparametersthatarenotdirectlytunablebycon-ventionaloptimizers(e.g.periodandamplitudeforsin,ninxn).LearningHLearningwOriginal Linear Model: w • HHFeature Inductionw’Induced Linear Model: w’ • H’H’Figure1:Top:Atraditionallearningprocedure,assign-ingasetofweightstoafixedfeatureset.Bottom:Dis-cretization,ourfeatureinductiontechnique,expandsthefeaturesetaspartoflearning,whilestillproducingalin-earmodelforinference,albeitwithmorefeatures.Discretizationallowsustoavoidmanynon-linearities(§2.2)whilepreservingthefastinferenceprovidedbyfeaturelocality(§2.1).Wefirstdis-cretizereal-valuedfeaturesintoasetofindicator5OnecouldalsoimagineafeaturetransformationfunctionΦthatreturnsavectorofbinsforasinglevaluereturnedbyafeaturefunctionhoratransformationthathasaccesstovaluesfrommultiplefeaturefunctionsatonce.
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featuresandthenuseaconventionaloptimizertolearnaweightforeachindicatorfeature(Figure1).Thistechniqueissometimesreferredtoasbinningandiscloselyrelatedtoquantization.Effectively,discretizationallowsustore-shapeafeaturefunc-tion(Figure2).Infact,givenaninfinitenumberofbins,wecanperformanynon-lineartransformationoftheoriginalfunction.1.00.5h01.00.5h1h2h3h4RFigure2:左边:Areal-valuedfeature.Bolddotsrepre-sentpointswherewecouldimaginebinsbeingplaced.However,sincewemayonlyadjustw0,these“bins”willberigidlyfixedalongthefeaturefunction’svalue.Right:Afterdiscretizingthefeatureinto4bins,wemaynowadjust4weightsindependently,toachieveanon-linearre-shapingofthefunction.Forindicatordiscretization,wedefineΦiintermsofabinningfunctionBINi(X)∈R→N:Φi(X)=h1,i_BINi(X)我(4)wherethe_operatorindicatesconcatenationofafeatureidentifierwithabinidentifiertoformanew,uniquefeatureidentifier.3.1LocalDiscretizationUnlikeotherapproachestonon-linearlearninginMT,weperformnon-lineartransformationonpar-tialhypothesesasinequation3wherediscretiza-tionisappliedasΦi(你好(d)),whichallowslocallynon-lineartransformations,insteadofapplyingΦtocompletehypothesesasinΦi(Hi(D)),whichwouldallowgloballynon-lineartransformations.Thisen-ablesourtransformedmodeltoproducenon-linearresponseswithregardtotheinitialfeaturesetHwhileinferenceremainslinearwithregardtotheoptimizedparametersw0.Importantly,ourtrans-formedfeaturesetrequiresnoadditionalnon-localinformationforinference.Byperformingtransformationswithinalocalcontext,weeffectivelyreinterpretthefeatureset.Forexample,thefamiliartargetwordcountfeaturefoundinmanymodernMTsystemsisoftenconcep-tualizedas“whatisthecountoftargetwordsinthecompletehypothesis?”Ahypothesis-levelviewofdiscretizationwouldviewthisas“Didthishypoth-esishave5targetwords?”.Onlyonesuchfeaturewillfireforeachhypothesis.However,localdis-cretizationreinterpretsthisfeatureas“Howmanyphrasesinthecompletehypothesishave1targetword?”Manysuchfeaturesarelikelytofireforeachhypothesis.WeprovideafurtherexampleofthistechniqueinFigure3.hTM=0.1hTM=0.2hCount=2hTM=0.113hTM_0.1=1hTM_0.2=1hTM_0.1=1hCount_2=1el gato come furtivamenteFigure3:Weperformdiscretizationlocallyoneachgrammarruleorphrasepair,operatingonthelocalfea-turevectorsh.Inthisexample,theoriginalreal-valuedfeaturesarecrossedoutwithasolidgraylineandtheirdiscretizedindicatorfeaturesarewrittenabove.Whenformingacompletehypothesisfrompartialhypotheses,wesumthecountsoftheseindicatorfeaturestoob-tainthecompletefeaturevectorH.Inthisexample,H={HTM0.1:2,HTM0.2:1,HCount2:1}Intermsofpredictivepower,thistransformationcanprovidethelearnedmodelwithincreasedabil-itytodiscriminatebetweenhypotheses.Thisispri-marilyaresultofmovingtoahigher-dimensionalfeaturespace.Asweintroducenewparameters,weexpectthatsomehypothesesthatwerepreviouslyin-distinguishableunderHbecomeseparableunderH0(§2.2).Weshowspecificexamplescomparinglin-ear,locallynon-linear,andgloballynon-linearmod-elsinFigures4-6.Asseenintheseexamples,lo-callynon-linearmodels(Eq.3,4)arenotanapprox-imationnorasubsetofgloballynon-linearmodels,butratheradifferentclassofmodels.3.2BinningAlgorithmToinitializethelearningprocedure,weconstructthebinningfunctionBINusedbytheindicatordi-
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LinearGloballyNon-LinearLocallyNon-LinearRanking∗S13heh:1.0saysh:1.0={H:2.0}S23sheh:2.0saidh:2.0={H:4.0}S14smallh:2.0kittenh:2.0={H:4.0}∗S24bigh:3.0lionh:3.0={H:6.0}∗S13heh:1.0saysh:1.0={H2:1}S23sheh:2.0saidh:2.0={H4:1}S14smallh:2.0kittenh:2.0={H4:1}∗S24bigh:3.0lionh:3.0={H6:1}∗S13heh1:1saysh1:1={H1:2}S23sheh2:1saidh2:1={H2:2}S14smallh2:1kittenh2:1={H2:2}∗S24bigh3:1lionh3:1={H3:2}Pairs(S13,S23){∆H:-2.0}⊕(S23,S13){∆H:2.0}(西德:9)(S24,S14){∆H:-2.0}(西德:9)(S14,S24){∆H:2.0}⊕(S13,S23){∆H2:1,∆H4:-1}⊕(S23,S13){∆H2:-1,∆H4:1}(西德:9)(S24,S14){∆H4:1,∆H6:-1}(西德:9)(S14,S24){∆H4:-1,∆H6:1}⊕(S13,S23){∆H1:2,∆H2:-2}⊕(S23,S13){∆H1:-2,∆H2:2}(西德:9)(S24,S14){∆H2:2,∆H3:-2}(西德:9)(S14,S24){∆H2:-2,∆H3:2}⊕PairwiseRankingΔH-202⊖⊕⊕⊖InseparableΔH2-11⊕-11ΔH4⊕H6:1H6:-1⊖⊖SeparableΔH1-11⊕-11ΔH2⊕H3:1H3:-1⊖⊖SeparableFigure4:AnexampleshowingacollinearityovermultipleinputsentencesS3,S4inwhichtheoracle-besthypothesisis“trapped”alongalinewithotherlowerqualityhypothesesinthelinearmodel’soutputspace.Rankingshowshowthehypotheseswouldappearinak-bestlistwitheachpartialderivationhavingitspartialfeaturevectorhunderit;thecompletefeaturevectorHisshowntotherightofeachhypothesisandtheoracle-besthypothesisisnotatedwitha∗.Pairsexplicatestheimplicitpairwiserankings.PairwiseRankinggraphsthosepairsinordertovisualizewhetherornotthehypothesesareseparable.(⊕indicatesthatthepairofhypothesesisrankedcorrectlyaccordingtotheextrinsicmetricand(西德:9)indicatesthepairisrankedincorrectly.Inthepairwiserankingrow,some⊕and(西德:9)pointsareannotatedwiththeirpositionsalongthethirdaxisH3(omittedforclarity).Collinearitycanalsooccurwithasingleinputhavingatleast3hypotheses.LinearGloballyNon-LinearLocallyNon-LinearRanking∗S12someh:2.0thingsh:2.0={H:4.0}S22somethingh:4.0={H:4.0}∗S12someh:2.0thingsh:2.0={H4:1}S22somethingh:4.0={H4:1}∗S12someh2:1thingsh2:1={H2:2}S22somethingh4:1={H4:1}Pairs(S12,S22){∆H:0.0}⊕(S22,S12){∆H:0.0}(西德:9)(S12,S22){∆H4:0}⊕(S22,S12){∆H4:0}(西德:9)(S12,S22){∆H2:2,∆H4:-1}⊕(S22,S12){∆H2:-2,∆H4:1}(西德:9)PairwiseRankingInseparableInseparableSeparableFigure5:Anexampleshowingatrivial“collision”inwhichtwohypothesesofdifferingqualityreceivethesamemodelscoreuntillocaldiscretizationisapplied.ThetwohypothesesareindistinguishableunderalinearmodelwiththefeaturesetH,asshownbythezero-differenceinthe“pairs”row.Whileagloballynon-lineartransformationdoesnotyieldanyimprovement,localdiscretizationallowsthehypothesestobeproperlyrankedduetothehigher-dimensionalfeaturespaceH2,H4.SeeFigure4foranexplanationofnotation.
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LinearLocallyNon-LinearRanking∗hB:1.0={HB:1.0}hA:1.0hA:1.0={HA:2.0}∗hA:1.0hA:1.0hB:1.0={HA:2.0,HB:1.0}hA:0.0={}∗hB:-4.0hB:1.0hB:1.0hB:1.0={HB:-1.0}hA:1.0={HA:1.0}∗hB:-4.0hA:1.0={HA:1.0,HB:-4.0}hB:1.0hB:1.0={HB:2.0}∗hB1:1={HB1:1}hA1:1hA1:1={HA1:2}∗hA1:1hA1:1hB1:1={HA1:2,HB1:1}hA1:0={}∗hB−4:1hB1:1hB1:1hB1:1={HB−4:1,HB1:3}hA1:1={HA1:1}∗hB−4:1hA1:1={HA1:1,HB−4:1}hB1:1hB1:1={HB1:2}PairwiseRankingΔHA-66⊕-66ΔHB⊖⊖⊖⊕⊕⊕⊖InseparableΔHA-33⊕-33ΔHB⊖⊖⊖⊕⊕⊕⊖11HB:1-4HB:-1-4HB:1-4HB:-1-4SeparableFigure6:Anexampledemonstratinganon-lineardecisionboundaryinducedbydiscretization.Thenon-linearnatureofthedecisionboundarycanbeseenclearlywhentheinducedfeaturesetHA1,HB1,HB−4(正确的)isconsideredintheoriginalfeaturespaceHA,HB(左边).Inthepairwiserankingrow,twoaxes(HA1,HB1)areplottedwhilethethirdaxisHB−4isindicatedonlyasstand-offannotationsforclarity.Givenalargernumberofhypotheses,suchsituationscouldalsoarisewithinasinglesentence.SeeFigure4foranexplanationofnotation.cretizerΦ.Wehavetwodesiderata:(1)anymono-tonictransformationofafeatureshouldnotaffecttheinducedbinningsinceweshouldnotrequirefeatureengineerstodeterminetheoptimalfeaturetransformationand(2)nobin’sdatashouldbesosparsethattheoptimizercannotreliablyestimateaweightforeachbin.Therefore,weconstructbinsthatare(我)populateduniformlysubjectto(二)eachbincontainingnomorethanonefeaturevalue.Wecallthisapproachuniformpopulationfeaturebin-ning.Whileonecouldconsiderthepredictivepowerofthefeatureswhendeterminingbinboundaries,thiswouldsuggestthatweshouldjointlyoptimizeanddeterminebinboundaries,whichisbeyondthescopeofthiswork.ThisproblemhasrecentlybeenconsideredforNLPbySuzukiandNagata(2013)andforMTbyLiuetal.(2013乙),thoughthelatterinvolvesdecodingtheentiretrainingdata.LetXbethelistoffeaturevaluestobinwhereiindexesfeaturevaluesxi∈Xandtheirassoci-atedfrequenciesfi.Wewanteachbintohaveauniformsizeu.Forthesakeofsimplifyingourfi-nalalgorithm,wefirstcreateadjustedfrequenciesf0isothatveryfrequentfeaturevalueswillnotoc-cupymorethan100%ofabinviathefollowingal-gorithm,whichiteratesoverk:uk=1|X||X|Xi=1fki(5)fk+1i=min(fki,uk)(6)whichreturnsu0=ukwhenfki
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