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 ID: 6971 Download Pdf

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

The Mistake Bound model In this lecture we study the online learning protocol In this setting the following scenario is repeated inde57356nitely 1 The algorithm receives an unlabeled example 2 The algorithm predicts a classi57356cation of this examp

Lecturer: . Yishay. . Mansour. Elad. . Walach. Alex . Roitenberg. Introduction. Up until . now, our algorithms start with . input and . work with it. suppose input arrives a little at a time, need instant .

Lecturer: . Yishay. . Mansour. Elad. . Walach. Alex . Roitenberg. Introduction. Up until . now, our algorithms start with . input and . work with it. suppose input arrives a little at a time, need instant .

1 MistakeBound Learning Mistakebound learning can be described in terms of playing an in64257nite learning game as follows 1 An adversary chooses some example and shows it to the learner 2 The learner tries to predict the label of the example 3 The

. Machine Learning. By:. WALEED ABDULWAHAB YAHYA AL-GOBI. MUHAMMAD BURHAN HAFEZ. KIM HYEONGCHEOL. HE RUIDAN. SHANG XINDI. . Overview. Introduction: . online learning vs. offline learning. Predicting from Expert Advice.

Kakade SKAKADE MICROSOFT COM Microsoft Research New England One Memorial Drive Cambridge MA 02142 USA Shai ShalevShwartz SHAIS CS HUJI AC IL School of Computer Science and Engineering The Hebrew University of Jerusalem Givat Ram Jerusalem 91904 Isra

Describing Inverse Problems. Syllabus. Lecture 01 Describing Inverse Problems. Lecture 02 Probability and Measurement Error, Part 1. Lecture 03 Probability and Measurement Error, Part 2 . Lecture 04 The L.

Inexact Theories. Syllabus. Lecture 01 Describing Inverse Problems. Lecture 02 Probability and Measurement Error, Part 1. Lecture 03 Probability and Measurement Error, Part 2 . Lecture 04 The L.

Lecture 02 . – . PAC Learning and tail bounds intro. CS 790-134 Spring 2015. Alex Berg. Today’s lecture. PAC Learning. Tail bounds…. Rectangle learning. +. -. -. -. -. -. -. +. +. +. Hypothesis .

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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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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

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