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1、MachineLearning10-701MidtermExamMarch4,2013Name:————————————————————————————————InstructionsAnythingonpaperisOKinarbitraryshapesize,andquantity.Electronicdevicesarenotacceptable.ThisincludesiPods,iPads,Androidtablets,Blackberries,Nokias,Windowsphones,MicrosoftSurfac
2、e,laptops,Chromebooks,MP3players,digitalcameras,GoogleGlass,oranythingelsethatrequireselectricity.1Ifweseeyouuseanysuchdevice,theexamisoverforyou.Yourresultswillnotcountandyouwillhavefailedthisexamautomatically,andtheremightmemoreseriousconsequencestoo.Noexceptions.S
3、witchoffyourphonesbeforetheexam.Therearemorequestionsintheexamthanwhatyouarelikelytobeabletosolvein70minutestime.Choosewiselyandanswerthose?rstthatyoubelieveyoucansolveeasily.Noneofthequestionsshouldrequiremorethan1pagetoanswer.It’sOKtobeconciseifyouaddressthekeypoi
4、ntsofthederivation(itshouldbehuman-readable,though).Grade:Problem1:/10Problem2:/5Problem3:/5Problem4:/5Problem5:/15Problem6:/5Problem7:/10Problem8:/5Problem9:/10Problem10:/10Problem11:/10Problem12:/10Total:/1001Obviousexceptionsforpacemakersandhearingaids.
5、1MachineLearning10-701MidtermExamMarch4,20131StochasticConvergenceProperties1.1Weakconvergenceisveryweak[5points]Provethattheconvergenceindistribution(a.k.aweakconvergence)doesnotimplythattheexpectedvaluesconvergeaswell.Inotherwords,constructanexamplesuchthatXn!DXbutE
6、[Xn]9E[X].Hint:Heavytails...1.2Strongconvergenceisstrongerthanweakconvergence[5points]Provethattheconvergenceindistributiondoesnotimplythealmostsurelyconvergence(a.k.a.strongconvergence).Inotherwords,constructanexamplesuchthatXn!DXbutXn9a:s:X.Hint:slidingwindows...2Ma
7、chineLearning10-701MidtermExamMarch4,20132GrowthfunctionandVCdimension[5points]LetSF(n)denotethegrowthfunctionandVCFbetheVapnik-ChevonenkisdimensionofafunctionsetF.UsingSauer’sLemmaprovethatS(n)(n+1)VCFF3MachineLearning10-701MidtermExamMarch4,20133Growthfunction[5poi
8、nts]LetF=ff:Rd!f
