Beyond Disagreementbased Agnostic Active Learning Chicheng Zhang University of California - PDF document

BeyondDisagreement-basedAgnosticActiveLearning ChichengZhangUniversityofCalifornia,SanDiego9500GilmanDrive,LaJolla,CA92093KamalikaChaudhuriUniversityofCalifornia,SanDiego9500GilmanDrive,LaJolla,CA92093Westudyagnosticactivelearning,wherethegoalistolearnaclassierinapre-edhypothesisclassinteractivelywithasfewlabelqueriesaspossible,whilemakingnoassumptionsonthetruefunctiongeneratingthelabels.Themainal-gorithmforthisproblemisdisagreement-basedactivelearning,whichhasahighlabelrequirement.Thusamajorchallengeisto�δndanalgorithmwhichachievesbetterlabelcomplexity,isconsistentinanagnosticsetting,andappliestogeneralcationproblems.Inthispaper,weprovidesuchanalgorithm.Oursolutionisbasedontwonovelcontributions;�δrst,areductionfromconsistentactivelearningtoconpredictionwithguaranteederror,andsecond,anovelcondence-ratedpredictor.1IntroductionInthispaper,westudyactivelearningofclassiersinanagnosticsetting,wherenoassumptionsaremadeonthetruefunctionthatgeneratesthelabels.Thelearnerhasaccesstoalargepoolofunlabelledexamples,andcaninteractivelyrequestlabelsforasmallsubsetofthese;thegoalistolearnanaccurateclassierinapre-speciedclasswithasfewlabelqueriesaspossible.Specically,wearegivenahypothesisclassHandatarget�δ,andouraimisto�δndabinaryclassierinHwhoseerrorisatmost�δmorethanthatofthebestclassierinH,whileminimizingthenumberofrequestedlabels.Therehasbeenalargebodyofpreviousworkonactivelearning;seethesurveysby[10,28]foroverviews.Themainchallengeinactivelearningisensuringconsistencyintheagnosticsettingwhilestillmaintaininglowlabelcomplexity.Inparticular,averynaturalapproachtoactivelearningistoviewitasageneralizationofbinarysearch[17,9,27].Whilethisstrategyhasbeenextendedtoseveraldifferentnoisemodels[23,27,26],itisgenerallyinconsistentintheagnosticcase[11].Theprimaryalgorithmforagnosticactivelearningiscalleddisagreement-basedactivelearningThemainideaisasfollows.AsetVofpossibleriskminimizersismaintainedwithtime,andthelabelofanexamplexisqueriedifthereexisttwohypotheseshhVsuchthath)�δ).Thisalgorithmisconsistentintheagnosticsetting[7,2,12,18,5,19,6,24];however,duetotheconservativelabelquerypolicy,itslabelrequirementishigh.Alineofworkdueto[3,4,1]haveprovidedalgorithmsthatachievebetterlabelcomplexityforlinearclassicationontheuniformdistributionovertheunitsphereaswellaslog-concavedistributions;however,theiralgorithmsarelimitedtothesespeciccases,anditisunclearhowtoapplythemmoregenerally.Thus,amajorchallengeintheagnosticactivelearningliteraturehasbeento�δndageneralactivelearningstrategythatappliestoanyhypothesisclassanddatadistribution,isconsistentintheagnos-ticcase,andhasabetterlabelrequirementthandisagreementbasedactivelearning.Thishasbeenmentionedasanopenproblembyseveralworks,suchas[2,10,4]. Inthispaper,weprovidesuchanalgorithm.Oursolutionisbasedontwokeycontributions,whichmaybeofindependentinterest.The�δrstisageneralconnectionbetweencondence-ratedpre-dictorsandactivelearning.Acondence-ratedpredictorisonethatisallowedtoabstainfrompredictiononoccasion,andasaresult,canguaranteeatargetpredictionerror.Givenaconratedpredictorwithguaranteederror,weshowhowtotoconstructanactivelabelqueryalgorithmconsistentintheagnosticsetting.Oursecondkeycontributionisanovelcondence-ratedpredictorwithguaranteederrorthatappliestoanygeneralclassicationproblem.Weshowthatourpredictoroptimalintherealizablecase,inthesensethatithasthelowestabstentionrateoutofallpredictorsguaranteeingacertainerror.Moreover,weshowhowtoextendourpredictortotheagnosticsetting.Combiningthelabelqueryalgorithmwithournovelcondence-ratedpredictor,wegetageneralactivelearningalgorithmconsistentintheagnosticsetting.Weprovideacharacterizationofthelabelcomplexityofouralgorithm,andshowthatthisisbetterthantheboundsknownfordisagreement-basedactivelearningingeneral.Finally,weshowthatforlinearclassicationwithrespecttotheuniformdistributionandlog-concavedistributions,ourboundsreducetothoseof[3,4].2Algorithm2.1TheSettingWestudyactivelearningforbinaryclassication.ExamplesbelongtoaninstancespaceX,andtheirlabelslieinalabelspaceY={1;labelledexamplesaredrawnfromanunderlyingdatadistributionDonX×Y.WeuseDtodenotethemarginalonDonX,andD|todenotetheconditionaldistributiononY|=xinducedbyD.Ouralgorithmhasaccesstoexamplesthroughtwooracles–anexampleoracleUwhichreturnsanunlabelledexamplex��XdrawnfromDalabellingoracleOwhichreturnsthelabelyofaninputx��XdrawnfromD|GivenahypothesisclassHofVCdimensiond,theerrorofanyh��Hwithrespecttoadatadistribution�δoverX×Yisdenedaserr)=Px,y)δ(x)�δy).Wede)=argminH),�κ)=err)).ForasetS,weabusenotationanduseStoalsodenotetheuniformdistributionovertheelementsofS.WedeP):=Px,y)δ),):=Ex,y)δ).GivenaccesstoexamplesfromadatadistributionDthroughanexampleoracleUandalabelingO,weaimtoprovideaclassi��Hsuchthatwithprobability��1�κ��,err��)+�δ,forsometargetvaluesof�δand��;thisisachievedinanadaptivemannerbymakingasfewqueriestothelabellingoracleOaspossible.When�κ)=0,wearesaidtobeintherealizablecase;inthemoregeneralagnosticcase,wemakenoassumptionsonthelabels,andthus)canbepositive.PreviousapproachestoagnosticactivelearninghavefrequentlyusedthenotionofdisagreementsThedisagreementbetweentwohypotheseshhwithrespecttoadatadistribution�δisthefractionofexamplesaccordingto�δtowhichhhassigndifferentlabels;formally:,h)=Px,y)δ)�δh)).Observethatadatadistribution�δinducesapseudo-��ontheelementsofH;thisiscalledthedisagreementmetric.Foranyrandanyh��H,Bh,r)tobethedisagreementballofradiusraroundhwithrespecttothedatadistribution.Formally:Bh,r)={H:��h,h��r}Fornotationalsimplicity,weassumethatthehypothesisspaceis“dense”withrepsecttothedatadistributionD,inthesensethat��r�0supB),r)h,h))=r.Ouranalysiswillstillapplywithoutthedensenessassumption,butwillbesignicantlymoremessy.Finally,givenasetofhypothesesV��H,thedisagreementregionofVisthesetofallexamplesxsuchthatthereexisttwohypothesesh,hVforwhichh)�δh).Thispaperestablishesaconnectionbetweenactivelearningandcondence-ratedpredictorswithguaranteederror.Acondence-ratedpredictorisapredictionalgorithmthatisoccasionallyal-lowedtoabstainfromclassication.Wewillconsidersuchpredictorsinthetransductivesetting.GivenasetVofcandidatehypotheses,anerrorguarantee��,andasetUofunlabelledexamples,acondence-ratedpredictorPeitherassignsalabelorabstainsfrompredictiononeachunlabelled ��U.ThelabelsareassignedwiththeguaranteethattheexpecteddisagreementbetweenthelabelassignedbyPandanyh��Vis����.Specically,forallh��V,PU(x)�δP(),P()�δ=0)����(1)ThisensuresthatifsomehVisthetrueriskminimizer,then,thelabelspredictedbyPonUdonotdifferverymuchfromthosepredictedbyh.Theperformanceofacondence-ratedpredictorwhichhasaguaranteesuchasinEquation(1)ismeasuredbyitscoverage,ortheprobabilityofPU()�δ=0);highercoverageimpliesbetterperformance.2.2MainAlgorithmOuractivelearningalgorithmproceedsinepochs,wherethegoalofepochkistoachieveexcessgeneralizationerror�δ�δ2+1,byqueryingafreshbatchoflabels.ThealgorithmmaintainsacandidatesetVthatisguaranteedtocontainthetrueriskminimizer.Thecriticaldecisionateachepochishowtoselectasubsetofunlabelledexampleswhoselabelsshouldbequeried.Wemakethisdecisionusingacondence-ratedpredictorP.Atepochk,werunwithcandidatehypothesissetV=Vanderrorguarantee��=�δ.WheneverPabstains,wequerythelabeloftheexample.Thenumberoflabelsmqueriedisadjustedsothatitisenoughtoachieveexcessgeneralizationerror�δ+1AnoutlineisdescribedinAlgorithm1;wenextdiscusseachindividualcomponentindetail. Algorithm1ActiveLearningAlgorithm:Outline ExampleoracleU,LabellingoracleO,hypothesisclassHofVCdimensionddence-ratedpredictorP,targetexcesserror�δandtargetcon��.k��log1.InitializecandidatesetVH.fork=12,..k�δ�δ+1�� k+1)UtogenerateafreshunlabelledsampleU{k,,...,zk,nofsizen512 ln192(512 +ln288 Runcondence-ratedpredictorPwithinpuyV=VU=Uanderrorguarantee=�δtogetabstentionprobabilities��k,,...,��k,nontheexamplesinU.Theseprobabilitiesinduceadistribution�κU.Let��PU()=0)= k,iintheRealizableCasethenm1536� ln1536� +ln .Drawmi.i.dexamplesfrom�κandqueryforlabelsoftheseexamplestogetalabelleddatasetS.UpdateV+1S+1{��Vh)=y,forall(x,y)��SInthenon-realizablecase,useAlgorithm2withinputshypothesissetV,distribution,targetexcesserror ,targetcon ,andthelabelingoracleOtogetanewhypothesissetV+1returnanarbitrary��V CandidateSets.Atepochk,wemaintainasetVofcandidatehypothesesguaranteedtocontainthetrueriskminimizerh)(w.h.p).Intherealizablecase,weuseaversionspaceasourcandidateset.TheversionspacewithrespecttoasetSoflabelledexamplesisthesetofallh��Hsuchthatx)=yforall(,y��S.Lemma1.SupposewerunAlgorithm1intherealizablecasewithinputsexampleoracleU,la-bellingoracleO,hypothesisclassH,condence-ratedpredictorP,targetexcesserror�δandtarget��.Then,withprobability1h)��Vforallk=12,...,k+1Inthenon-realizablecase,theversionspaceisusuallyempty;weuseinsteada(1�κ��)setforthetrueriskminimizer.GivenasetSofnlabelledexamples,letC()��Hbeafunctionof wheretheexpectationiswithrespecttotherandomchoicesmadebyP ;C()issaidtobea(1�κ��)dencesetforthetrueriskminimizerifforalldatadistributionsoverX×Y,�δκ,hκ(�)��C()]��1�κ��,Recallthath)=argminH).Inthenon-realizablecase,ourcandidatesetsare(1�κ��)dencesetsforh),for��=��.TheprecisesettingofVisexplainedinAlgorithm2.Lemma2.SupposewerunAlgorithm1inthenon-realizablecasewithinputsexampleoracleU,labellingoracleO,hypothesisclassH,condence-ratedpredictorP,targetexcesserror�δandtargetcon��.Thenwithprobability1�κ��,h)��Vforallk=12,...,k+1LabelQuery.Wenextdiscussourlabelqueryprocedure–whichexamplesshouldwequerylabelsfor,andhowmanylabelsshouldwequeryateachepoch?WhichLabelstoQuery?Ourgoalistoquerythelabelsofthemostinformativeexamples.Tochoosetheseexampleswhilestillmaintainingconsistency,weuseacondence-ratedpredictorPwithguaranteederror.TheinputstothepredictorareourcandidatehypothesissetVwhichcontains(w.h.p)thetrueriskminimizer,afreshsetUofunlabelledexamples,andanerrorguarantee��=.Fornotationsimplicity,assumetheelementsinUaredistinct.Theoutputisasequenceofabstentionprobabilities{k,��k,,...,��k,n,foreachexampleinU.Itinducesadistribution�κoverU,fromwhichweindependentlydrawexamplesforlabelqueries.HowManyLabelstoQuery?Thegoalofepochkistoachieveexcessgeneralizationerror�δToachievethis,passivelearningrequires(d/δlabelledexamplesintherealizablecase,and((κ)+�δexamplesintheagnosticcase.Akeyobservationinthispaperisthatinordertoachieveexcessgeneralizationerror�δD,itsuf�δcestoachieveamuchlargerexcessgeneralizationerrorO(onthedatadistributioninducedby�κD|,where��isthefractionofexamplesonwhichthecondence-ratedpredictorabstains.Intherealizablecase,weachievethisbysamplingm1536� ln1536� +ln i.i.dexamples�κ,andqueryingtheirlabelstogetalabelleddatasetS.Observethatas��istheabstentionprobabilityofPwithguaranteederror���δ,itisgenerallysmallerthanthemeasureofthedisagreementregionoftheversionspace;thiskeyfactresultsinimprovedlabelcomplexityoverdisagreement-basedactivelearning.Thissamplingprocedurehasthefollowingproperty:Lemma3.SupposewerunAlgorithm1intherealizablecasewithinputsexampleoracleU,la-bellingoracleO,hypothesisclassH,condence-ratedpredictorP,targetexcesserror�δandtarget��.Thenwithprobability1�κ��,forallk=12,...,k+1,andforallh��V)���δ.Inparticular,thereturnedattheendofthealgorithmsatiseserr���δTheagnosticcasehasanaddedcomplication–inpractice,thevalueof�κisnotknownaheadoftime.Inspiredby[24],weuseadoublingprocedure(statedinAlgorithm2)whichadaptively�δthenumbermoflabelledexamplestobequeriedandqueriesthem.Thefollowingtwolemmasillustrateitsproperties–thatitisconsistent,andthatitdoesnotusetoomanylabelqueries.Lemma4.SupposewerunAlgorithm2withinputshypothesissetV,exampledistribution��labellingoracleO,targetexcesserror˜�δandtargetcon.LetbethejointdistributiononX×Yinducedby��andD|.Thenthereexistsanevent,P)��1�κ,suchthaton,(1)Algorithm2haltsand(2)thesetVhasthefollowingproperties:(2.1)Ifforh��H,err)�κerr))��˜�δ,thenh��V(2.2)Ontheotherhand,ifh��V,thenerr)�κerr))��˜�δWheneventhappens,wesayAlgorithm2succeeds.Lemma5.SupposewerunAlgorithm2withinputshypothesissetV,exampledistribution��labellingoracleO,targetexcesserror˜�δandtargetcon.Thereexistssomeabsoluteconstant0,suchthatontheeventthatAlgorithm2succeeds,ncdln �κ+ln )+˜�κ �κ.Thusthetotalnumberoflabelsqueriedis=122dln �κ+ln )+˜�κ �κ (hideslogarithmicfactors Anaiveapproach(seeAlgorithm4intheAppendix)whichusesanadditiveVCboundgivesasamplecomplexityofO((dln(1˜�δ)+ln(1))˜�δ;Algorithm2givesabettersamplecomplexity.Thefollowinglemmaisaconsequenceofourlabelqueryprocedureinthenon-realizablecase.Lemma6.SupposewerunAlgorithm1inthenon-realizablecasewithinputsexampleoracleU,labellingoracleO,hypothesisclassH,condence-ratedpredictorP,targetexcesserror�δandtargetcon��.Thenwithprobability1�κ��,forallk=12,...,k+1,andforallh��V)��err))+�δ.Inparticular,thereturnedattheendofthealgorithmsatis��err))+�δ Algorithm2AnAdaptiveAlgorithmforLabelQueryGivenTargetExcessError HypothesissetVofVCdimensiond,Exampledistribution��,LabelingoracleO,targetexcesserror˜�δ,targetcon.forj=12,...doDrawn=2i.i.dexamplesfrom��;querytheirlabelsfromOtogetalabelleddataset.Denote/j(j+1)).TrainanERMclassiVoverSnethesetVasfollows:��V:err)��err)+�δ ��(n)+ (nh,��(n,��):= ln +ln supV(n)+ (nh,���κ j,breakreturnV 2.3Condence-RatedPredictorOuractivelearningalgorithmusesacondence-ratedpredictorwithguaranteederrortomakeitslabelquerydecisions.Inthissection,weprovideanovelcondence-ratedpredictorwithguaranteederror.Thispredictorhasoptimalcoverageintherealizablecase,andmaybeofindependentinterest.ThepredictorPreceivesasinputasetV��Hofhypotheses(whichislikelytocontainthetrueriskminimizer),anerrorguarantee��,andasetofUofunlabelledexamples.Weconsiderasoftpredictionalgorithm;so,foreachexampleinU,thepredictorPoutputsthreeprobabilitiesthataddupto1–theprobabilityofpredicting1�κand0.Thisoutputissubjecttotheconstraintthattheexpecteddisagreementbetweenthe±labelsassignedbyPandthoseassignedbyanyh��Visatmost��,andthegoalistomaximizethecoverage,ortheexpectedfractionofnon-abstentions.Ourkeyinsightisthatthisproblemcanbewrittenasalinearprogram,whichisdescribedinAlgo-rithm3.Therearethreevariables,�≡�√��,foreachunlabelledzU;therearetheprobabil-itieswithwhichwepredict1�κ1and0onzrespectively.Constraint(2)ensuresthattheexpecteddisagreementbetweenthelabelpredictedandanyh��Visnomorethan��,whiletheLPobjectivemaximizesthecoverageundertheseconstraints.ObservethattheLPisalwaysfeasible.AlthoughtheLPhasinnitelymanyconstraints,thenumberofconstraintsinEquation(2)distinguishablebyisatmost(em/d),wheredistheVCdimensionofthehypothesisclassH.Theperformanceofacondence-ratedpredictorismeasuredbyitserrorandcoverage.Theerrorofacondence-ratedpredictoristheprobabilitywithwhichitpredictsthewronglabelonanexample,whilethecoverageisitsprobabilityofnon-abstention.WecanshowthefollowingguaranteeontheperformanceofthepredictorinAlgorithm3.Theorem1.Intherealizablecase,ifthehypothesissetVistheversionspacewithrespecttoatrainingset,thenPU()�δh),P()�δ=0)����.Inthenon-realizablecase,ifthehypothesissetVisan(1�κ��)dencesetforthetrueriskminimizerh,then,w.p��1�κ��,U()�δy,P(x)�δ=0)��PU)�δy)+��. wheretheexpectationistakenovertherandomchoicesmadebyP Algorithm3Condence-ratedPredictor hypothesissetV,unlabelleddataU={,...,z,errorbound��.Solvethelinearprogram:minsubjectto:��i,�≡�√��=1��V,(z(z1��m(2)i,�≡�√��0ForeachzU,outputprobabilitiesforpredicting1�κand0�≡�√,and�� Intherealizablecase,wecanalsoshowthatourcondenceratedpredictorhasoptimalcoverage.Observethatwecannotdirectlyshowoptimalityinthenon-realizablecase,astheperformancedependsontheexactchoiceofthe(1�κ��)-conδdenceset.Theorem2.Intherealizablecase,supposethatthehypothesissetVistheversionspacewithrespecttoatrainingset.IfPisanycondenceratedpredictorwitherrorguarantee��,andifPisthepredictorinAlgorithm3,then,thecoverageofPisatleastmuchasthecoverageofP3PerformanceGuaranteesAnessentialpropertyofanyactivelearningalgorithmisconsistency–thatitconvergestothetrueriskminimizergivenenoughlabelledexamples.Weobservethatouralgorithmisconsistentpro-videdweuseanycondence-ratedpredictorPwithguaranteederrorasasubroutine.Theconsis-tencyofouralgorithmisaconsequenceofLemmas3and6andisshowninTheorem3.Theorem3(Consistency)SupposewerunAlgorithm1withinputsexampleoracleU,labellingoracleO,hypothesisclassH,condence-ratedpredictorP,targetexcesserror�δandtarget��.Thenwithprobability1�κ��,theclassireturnedbyAlgorithm1satis�κerr))���δWenowestablishalabelcomplexityboundforouralgorithm;however,thislabelcomplexityboundappliesonlyifweusethepredictordescribedinAlgorithm3asasubroutine.ForanyhypothesissetV,datadistributionD,and��,de�δV,��)tobetheminimumabsten-tionprobabilityofacondence-ratedpredictorwhichguaranteesthatthedisagreementbetweenitspredictedlabelsandanyh��VunderDisatmost��.Formally,�δV,��)=min{E(x):EEI((x)=+1)√(x)+I(h(x)=�κ≡(x)]��forallh��V,��(x)+�≡()+�√()�≡1��(x),�≡(),�√()��0.De��(r,��):=,r),��).Thelabelcomplexityofouractivelearningalgorithmcanbestatedasfollows.Theorem4(LabelComplexity)SupposewerunAlgorithm1withinputsexampleoracleU,la-bellingoracleO,hypothesisclassH,condence-ratedpredictorPofAlgorithm3,targetexcesserror�δandtargetcon��.Thenthereexistconstantsc,c0suchthatwithprobability�κ��:(1)Intherealizablecase,thetotalnumberoflabelsqueriedbyAlgorithm1isatmost:log =1lnδ�δ +ln(log(1/δ)��κk+1 δ�δ (2)Intheagnosticcase,thetotalnumberoflabelsqueriedbyAlgorithm1isatmost:log =1ln(2κ)+�δ�δ +ln(log(1/δ)��κk+1 (2κ)+�δ�δ ) k)6 Thelabelcomplexityofdisagreement-basedactivelearningischaracterizedintermsofthedisagreementcoef�δ.Givenaradiusr,thedisagreementcoef�δ�θ(r)isde(r)=supDIS(B,r))) whereforanyV��H,DISV)isthedisagreementregionofV.AsPDIS(B,r)))=r,0)[13],inournotation,�θ(r)=supr Intherealizablecase,thebestknownboundforlabelcomplexityofdisagreement-basedactivelearningis(()·ln(1·(dln�θ()+lnln(1/δ)))[20].Ourlabelcomplexityboundmaybeedto: sup�आlog(1/κ)�δ�δ lnsup�आlog(1/κ)�δ�δ +lnln whichisessentiallytheboundof[20]with�θ()replacedbysup�आlog(1/κ)�κ256) .Asen-forcingalowererrorguaranteerequiresmoreabstention,��r,��)isadecreasingfunctionof��;asasup�आlog(1/κ)�δ�δ �θ(),andourlabelcomplexityboundisbetter.Intheagnosticcase,[12]providesalabelcomplexityboundof((2)+�δ)·() ln(1/δ)+lnδ)))fordisagreement-basedactive-learning.Incontrast,byProposition1ourlabelcom-plexityisatmost:sup�आlog(1/κ)�(2κ)+�δ�δ )+�δ) ln(1/δ)+dlnδ)Again,thisisessentiallytheboundof[12]with�θ(2κ)+�δreplacedbythesmallerquantitysup�आlog(1/κ)�(2κ)+�δ�δ )+�δ[20]hasprovidedamorerenedanalysisofdisagreement-basedactivelearningthatgivesalabelcomplexityof(()+�δ)() +ln dln�θ(κ)+�δ)+lnln ;observethattheirdependenceisstillon�θ(κ)+�δ).Weleaveamorerenedlabelcomplexityanalysisofouralgorithmforfuturework.Animportantsub-caseoflearningfromnoisydataislearningundertheTsybakovnoisecondi-tions[30].WedeferthediscussionintotheAppendix.3.1CaseStudy:LinearClassicationundertheLog-concaveDistributionWenowconsiderlearninglinearclassierswithrespecttolog-concavedatadistributiononR.Inthiscase,foranyr,thedisagreementcoef�δ�θ(r)��O( ln(1))[4];however,forany���0r,) O(ln(r/))(seeLemma14intheAppendix),whichismuchsmallersolongas��/risnottoosmall.Thisleadstothefollowinglabelcomplexitybounds.Corollary1.SupposeDisisotropicandlog-concaveonR,andHisthesetofhomogeneouslin-earclassiersonR.ThenAlgorithm1withinputsexampleoracleU,labellingoracleO,hypothesisH,condence-ratedpredictorPofAlgorithm3,targetexcesserror�δandtargetcon��esthefollowingproperties.Withprobability1�κ��:(1)Intherealizablecase,thereexistssomeabsoluteconstantc0suchthatthetotalnumberoflabelsqueriedisatmostc +lnln +ln ). Herethe(notationhidesfactorslogarithmicin1/δ (2)Intheagnosticcase,thereexistssomeabsoluteconstantc0suchthatthetotalnumberofla-belsqueriedisatmostc) +ln )ln) ln) +ln )+ln ) lnln (3)If(C�κ-TsybakovNoiseconditionholdsforDwithrespecttoH,thenthereexistssomec0(thatdependsonC�κ)suchthatthetotalnumberoflabelsqueriedisatmost 2ln1 ln +ln Intherealizablecase,ourboundmatches[4].Fordisagreement-basedalgorithms,theboundis( 2ln21 (lnd+lnln ,whichisworsebyafactorofO( ln(1)).[4]doesnotaddressthefullyagnosticcasedirectly;however,if�κ)isknowna-priori,thentheiralgorithmcanachieveroughlythesamelabelcomplexityasours.FortheTsybakovNoiseConditionwith�κκ1,[3,4]providesalabelcomplexityboundfor( 2ln21 +lnln withanalgorithmthathasa-prioriknowledgeofC�κ.Wegetaslightlybetterbound.Ontheotherhand,adisagreementbasedalgorithm[20]givesalabelcomplexityof( 2ln21 2 (lnd+lnln .Againourboundisbetterbyfactorof�� overdisagreement-basedalgorithms.For�κ=1,wecantightenourlabelcomplexitytogeta(ln +lnln +ln bound,whichagainmatches[4],andisbetterthantheonesprovidedbydisagreement-basedalgorithm–( 2ln21 (lnd+lnln [20].4RelatedWorkActivelearninghasseenalotofprogressoverthepasttwodecades,motivatedbyvastamountsofunlabelleddataandthehighcostofannotation[28,10,20].Accordingto[10],thetwomainthreadsofresearchareexploitationofclusterstructure[31,11],andef�δcientsearchinhypothesisspace,whichisthesettingofourwork.WearegivenahypothesisclassH,andthegoalisto�δndanh��Hthatachievesatargetexcessgeneralizationerror,whileminimizingthenumberoflabelqueries.Threemainapproacheshavebeenstudiedinthissetting.The�δrstandmostnaturaloneisgeneralizedbinarysearch[17,8,9,27],whichwasanalyzedintherealizablecaseby[9]andinvariouslimitednoisesettingsby[23,27,26].Whilethisapproachhastheadvantageoflowlabelcomplexity,itisgenerallyinconsistentinthefullyagnosticsetting[11].Thesecondapproach,disagreement-basedactivelearning,isconsistentintheagnosticPACmodel.[7]providesthe�δrstdisagreement-basedalgorithmfortherealizablecase.[2]providesanagnosticdisagreement-basedalgorithm,whichisanalyzedin[18]usingthenotionofdisagreementcoef�δcient.[12]reducesdisagreement-basedactivelearningtopassivelearning;[5]and[6]furtherextendthisworktoprovidepracticalandef�δcientimplementations.[19,24]givealgorithmsthatareadaptivetotheTsybakovNoisecondition.Thethirdlineofwork[3,4,1],achievesabetterlabelcomplexitythandisagreement-basedactivelearningforlinearclassiersontheuniformdistributionoverunitsphereandlogconcavedistribu-tions.However,alimitationisthattheiralgorithmappliesonlytothesespecicsettings,anditisnotapparenthowtoapplyitgenerally.Researchoncondence-ratedpredictionhasbeenmostlyfocusedonempiricalwork,withrelativelylesstheoreticaldevelopment.TheoreticalworkonthistopicincludesKWIKlearning[25],confor-malprediction[29]andtheweightedmajorityalgorithmof[16].Theclosesttoourworkistherecentlearning-theoretictreatmentby[13,14].[13]addressescondence-ratedpredictionwithguaranteederrorintherealizablecase,andprovidesapredictorthatabstainsinthedisagreementregionoftheversionspace.Thispredictorachieveszeroerror,andcoverageequaltothemeasureoftheagree-mentregion.[14]showshowtoextendthisalgorithmtothenon-realizablecaseandobtainzeroerrorwithrespecttothebesthypothesisinH.Notethatthepredictorsin[13,14]generallyachievelesscoveragethanoursforthesameerrorguarantee;infact,ifweplugthemintoourAlgorithm1,thenwerecoverthelabelcomplexityboundsofdisagreement-basedalgorithms[12,19,24].Aformalconnectionbetweendisagreement-basedactivelearninginrealizablecaseandperfectdence-ratedprediction(withazeroerrorguarantee)wasestablishedby[15].Ourworkcanbeseenasasteptowardsbridgingthesetwoareas,bydemonstratingthatactivelearningcanbefurtherreducedtoimperfectcondence-ratedprediction,withpotentiallyhigherlabelsavings.Acknowledgements.WethankNSFunderIIS-1162581forresearchsupport.WethankSanjoyDasguptaandYoavFreundforhelpfuldiscussions.CZwouldliketothankLiweiWangforintro-ducingtheproblemofselectiveclassicationtohim. 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