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1、JournalofMachineLearningResearch13(2012)2655-2684Submitted4/11;Revised2/12;Published9/12CoherenceFunctionswithApplicationsinLarge-MarginClassi?cationMethodsZhihuaZhangZHZHANG@ZJU.EDU.CNDehuaLiuDEHUALIU0427@GMAIL.COMGuangDaiGUANG.GDAI@GMAIL.COMCollegeofComput
2、erScienceandTechnologyZhejiangUniversityHangzhou,Zhejiang310027,ChinaMichaelI.JordanJORDAN@CS.BERKELEY.EDUComputerScienceDivisionandDepartmentofStatisticsUniversityofCaliforniaBerkeley,CA94720-1776,USAEditor:XiaotongShenAbstractSupportvectormachines(SVMs)nat
3、urallyembodysparsenessduetotheiruseofhingelossfunc-tions.However,SVMscannotdirectlyestimateconditionalclassprobabilities.Inthispaperweproposeandstudyafamilyofcoherencefunctions,whichareconvexanddifferentiable,assur-rogatesofthehingefunction.Thecoherencefunct
4、ionisderivedbyusingthemaximum-entropyprincipleandischaracterizedbyatemperatureparameter.Itbridgesthehingefunctionandthelogitfunctioninlogisticregression.Thelimitofthecoherencefunctionatzerotemperaturecorrespondstothehingefunction,andthelimitoftheminimizerofi
5、tsexpectederroristheminimizeroftheexpectederrorofthehingeloss.Werefertotheuseofthecoherencefunctioninlarge-marginclas-si?cationasC-learning,andwepresentef?cientcoordinatedescentalgorithmsforthetrainingofregularizedC-learningmodels.Keywords:large-marginclassi
6、?ers,hingefunctions,logisticfunctions,coherencefunctions,C-learning1.IntroductionLarge-marginclassi?cationmethodshavebecomeincreasinglypopularsincetheadventofboost-ing(Freund,1995),supportvectormachines(SVM)(Vapnik,1998)andtheirvariantssuchasψ-learning(Shene
7、tal.,2003).Large-marginclassi?cationmethodsaretypicallydevisedbasedonamajorization-minimizationprocedure,whichapproximatelysolvesanotherwiseintractableopti-mizationproblemde?nedwiththe0-1loss.Forexample,theconventionalSVMemploysahingeloss,theAdaBoostalgorith
8、memploystheexponentialloss,andψ-learningemploysaso-calledψ-loss,asmajorizationsofthe0-1loss.Large-marginclassi?cationmethodscanbeuni?edusingthetoolsofregularizationtheory;thatis,theycanbeexpress
