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CMSC Spring Learning Theory Lecture Mistake Bound Model Halving Algorithm Linear Classiers Instructors Sham Kakade and Ambuj Tewari Introduction This course will be divided into parts

In each part we will make different assumptions about the data generating process Online Learning No assumptions about data generating process Worst case analysis Fundamental connections to Game Theory Statistical Learning Assume data consists of in

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CMSC Spring Learning Theory Lecture Mistake Bound Model Halving Algorithm Linear Classiers Instructors Sham Kakade and Ambuj Tewari Introduction This course will be divided into parts




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Presentation on theme: "CMSC Spring Learning Theory Lecture Mistake Bound Model Halving Algorithm Linear Classiers Instructors Sham Kakade and Ambuj Tewari Introduction This course will be divided into parts"— Presentation transcript:

Notethatweareignoringefciencyissueshere.WehavenotsaidanythingabouttheamountofcomputationAhastodoineachroundinordertoupdateitshypothesisfromhttoht+1.Settingthisissueasideforamoment,wehavearemarkablysimplealgorithmHALVING(C)thathasamistakeboundoflg(jCj)foranyniteconceptclassC.ForanitesetHofhypotheses,denethehypothesismajority(H)asfollows,majority(H)(x):=(+1jfh2Hjh(x)=+1gjjHj=2;�1otherwise: Algorithm1HALVING(C) C1 Ch1 majority(C1)fort=1toTdoReceivextPredictht(xt)ReceiveytCt+1 ff2Ctjf(xt)=ytght+1 majority(Ct+1)endfor Theorem2.2.ForanyniteconceptclassC,wehavemistake(HALVING(C);C))lgjCj:Proof.ThekeyideaisthatifthealgorithmmakesamistakethenatleasthalfofthehypothesisinCtareeliminated.Formally,ht(xt)6=yt)jCt+1jjCtj=2:Therefore,denotingthenumberofmistakesuptotimetbyMt,Mt:=TXt=11[ht(xt)6=yt];wehavejCt+1jjC1j 2Mt=jCj 2Mt(1)Sincethereisanf2Cwhichperfectlyclassiesallxt,wealsohave1jCt+1j:(2)Combining(1)and(2),wehave1jCj 2Mt;whichgivesMtlg(jCj). 3LinearClassiersandMarginLetusnowlookataconcreteexampleofaconceptclass.SupposeX=Rdandwehaveavectorw2Rd.Wedenethehypothesis,hw(x)=sgn(wx);2 Thisboundisnicebecauseeventhoughwehadanuncountableconceptclasstobeginwith,themarginassumptionallowedustoworkwithanitesubsetoftheconceptclassandwewereabletoderiveamistakebound.However,theresultisunsatisfactorybecauserunningthehalvingalgorithmonC linisextremelyinefcient.Onemightwonderifonecanusethespecialstructureofthespaceoflinearclassierstoimplementthehalvingalgorithmmoreefciently.Indeed,itpossibletoimplementavariantofthehalvingalgorithmefcientlyusingtheellipsoidmethoddevelopedforthelinearprogrammingfeasibilityproblem.Notethatthemistakebounddependsexplicitlyonthedimensiondoftheproblem.Wewouldalsoliketobeabletogiveadimensionindependentmistakebound.Indeed,aclassicalgorithmcalledPERCEPTRONhassuchamistakebound.4