rademacher and gaussian complexities risk bounds and structural results

rademacher and gaussian complexities risk bounds and structural results

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时间:2019-03-01

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1、Radema herandGaussianComplexities:RiskBoundsandStru turalResults12eter.BartlettandShaharendelson1BwulfTe hnologies2030AddisonStreet,Suite102Berkeley,CA94704,USAbartlettbarnhillte hnologies. om2Resear hS hoolofnformationS ien esandEngineeringAustralianationalUnivers

2、ityCanberra0200,Australiashahar sl.anu.edu.auAbstra t.Weinvestigatetheuseof ertaindata-dependentestimatesofthe omplexityofafun tion lass, alledRadema herandgaussian omplexities.nade isiontheoreti setting,weprovegeneralriskboundsintermsofthese omplexities.We onsiderfun ti

3、on lassesthat anbeexpressedas ombinationsoffun tionsfrombasis lassesandshowhowtheRadema herandgaussian omplexitiesofsu hafun tion lass anbeboundedintermsofthe omplexityofthebasis lasses.Wegiveexam-plesoftheappli ationofthesete hniquesin ndingdata-dependentriskboundsforde i

4、siontrees,neuralnetworksandsupportve torma hines.1ntrodu tionnlearningproblemslikepattern lassi ationandregression,a onsiderableamountofe orthasbeenspentonobtaininggooderrorbounds.Theseareuseful,forexample,fortheproblemofmodelsele tion

5、 hoosingamodelofsuitable omplexity

6、.Typi ally,su hboundstaketheformofasumoftwoterms:somesample-basedestimateofperforman eandapenaltytermthatislargeformore omplexmodels.Forexample,inpattern lassi ation,thefollowingtheoremisanimprovementofa lassi alresultofVapnikandChervonenkis[20℄.Theorem1.etFbea lassoff1

7、g-valuedfun tionsde nedonasetX.etbeaprobabilitydistributiononXf1g,andsupposethat(X;Y);:::;(X;Y)11nnand(X;Y)are hosenindependentlya ordingto.Then,thereisanabsoluteonstantsu hthatforanyintegern,withprobabilityatleast1Æoversamplesoflengthn,everyfinFsatis esrVCdim(F)^(Y

8、6=f(X))(Y6=f(X))+;nn^whereVCdim(F)denotestheVapnik-ChervonenkisdimensionofF,(S)=nn(1=n)1(X;Y),and1istheindi atorfun tionofS,SiiSi=12eterBartlettandShaharendelsonnthis ase,thesample-basedestimateofperforman eistheproportionofexamplesinthetrainingsamplethataremis lass

9、i edbythefun tionf,andthe omplexitypenaltyterminvolvestheVC-dimensionofthe lassoffun tion

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