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首页 Statistical Learning Theory Vapnik.pdf

Statistical Learning Theory Vapnik.pdf

Statistical Learning Theory Vap…

上传者: 鳴开朗奇罗 2017-07-24 评分 0 0 0 0 0 0 暂无简介 简介 举报

简介:本文档为《Statistical Learning Theory Vapnikpdf》,可适用于人文社科领域,主题内容包含AdaptiveandLearningSystemsforSignalProcessing,Communications,andControlEdt符等。

AdaptiveandLearningSystemsforSignalProcessing,Communications,andControlEdto~:SimonHuykinWerbosTHEROOTSOFBACKPROPAGATION:FromOrderedDerivativestoNeuralNetworksandpoliticalForecastingKrshcKanellakopoulos,andKokotovic:NONLINEARANDADAPTIVECONTROLDESIGNNiluasandShaoSIGNALPROCESSINGWITHALPHASTABLEDISTRIBUTIONSANDAPPLICATIONSDiamantarasandKungPRINCIPALCOMPONENTNEURALNETWORKS:THEORYANDAPPLICATIONSTaoandKokotovic:ADAPTIVECONTROLOFSYSTEMSWITHACTUATORANDSENSORNONLINEARITIESTsoukalasFUZZYANDNEURALAPPROACHESINENGINEERINGHrycejNEUROCONTROL:TOWARDSANINDUSTRIALCONTROLMETHODOLOGYBeckemanADAPTIVECOOPERATIVESYSTEMSCherkasskyandMuherLEARNINGFROMDATA:CONCEPTS,THEORY,ANDMETHODSpassistoandBurgessSTABILITYANALYSISOFDISCRETEEVENTSYSTEMSSinchez~eiiaandSznaierROBUSTSYSTEMSTHEORYANDAPPLICATIONSVapnikSTATISTICALLEARNINGTHEORYStatisticalLearningTheoryAWtLEYINfERSCIENCEPUBLICATfONJOHNWLLfYSONS,LNC,NEWYORKCCilCHESTERjWEfNHEIMIBRISBANEjStNGAPOREITORONTOPageivDisclaimer:ThsbookcontainscharacterswithchacnticsWhenthecharacterscanberepresentedusingtheIScharacterset(http,netLibrarywillrepresentthemastheyappearintheorignaltext,andmostcomputerswillbeabletoshowthefullcharacterscorrectlyInordertokeepthetextsearchableandreadableonmostcomputers,characterswithhacriticsthatarenotpartoftheISlistwillberepresentedwithouttheirchacriticalmarksThsbookisprintedonacidfieepaper#CopynghtObyJohnWileySons,IncAllrightsreservedpublishedsimultaneouslyinCanadaNopartofth~spublicationmaybereproduced,storedinaretrievalsystemortransmittedinanyformorbyanymeans,electronic,mechanical,photocopyng,recordmg,scanningorotherwise,exceptaspermittedunderSectionsoroftheUnitedStatesCopyrightAct,withouteitherthepriorwrittenpermissionofthepublisher,orauthorizahonthroughpaymentoftheappropriatepercopyfeetotheCopynghtClearanceCenter,RosewoodDrive,Danvers,MA,(),fax()RequeststothepublisherforpermissionshouldbeaddressedtothepermissionsDepartment,JohnWileySons,Inc,ThrdAvenue,NewYork,NY,(),fax(),EMail:pERMREQWILEYCOMLibrmyofCongressCataloginginPublieutionData:Vapnik,VlahrNaurnovichStatisticallearningtheoryVladmirNVapnikpcm(Adaptiveandlearningsystemsforsignalprocessing,communications,andcontrol)IncludesbibliographcalreferencesandindexISBN(cloth:alkpaper)ComputationallearningtheoryITitleSeriesQVdcCIPprintedintheUnitedStatesofAmericaPagevInmemoryofmyfatherPagevi~CONTENTSPrefaceIntroduction:TheProhlen~ofInductionandStatidicalInferenceILearningParadigminStatisticsTwoApproachestoStatisticalInferenceParticular(I)arametncInference)andGeneral(NonparametncInference)TheParadigmCreatedbytheParametricApproachShortcomingoftheParametricParadigmAftertheClassicalParahgmTheRenassanceTheGeneralizationoftheGl~venkoCantelllKolmogorovTheoryTheStructuralR~skM~nimizationPnncipleTheManPnncipleofInferencefromaSmdSampleSlzeWhatThlsBookisAboutITheoryofLearningandGeneralizationTwoApproachestotheLeamlngProblemIGeneralModelofLearningfromExamplesTheProblemofhfinimizlngtheRiskFunchondfromEmpincdDataTheProblemofPatternRecognitionTheProblemofRegressionEstimationProblemofInterpretingResultsofIndirectMeasuringTheProblemofEensityEstimatlon(theFisherWaldSetting)InductionPrinciplesforMinimizingtheRlskFunctionalontheBasisofEmpiricalDataClassicalMethodsforSolvingtheFunctionEstimationProblemsIdentlficatlonofStochasticObjectsEst~mationoftheDensitiesandConditionalDensitiesProblemofDens~tyEst~mationDlrectSettingProblemofCondihanalProbabllltyEst~mahonProblemofConditionalDens~tyEstimationTheProblemofSolvinganApproximatelyDeterminedIntegralEquationIGhvenkoCanteltiTheoremConvergenceinProbabilityandAlmostSureConvergenceGhvenkoCantelThealemThreeImportantStatist~calLawsnlPosedProblemsTheStructureoftheLearnlngTheoryAppendixtoChapterMethodsforSolvingIPosedProblemsAlITheProblemofSolvlnganOperatorEquationAProblemsWellPased~nTlkhanovsSenseAlTheRegulx~zationMethodAlIdeaofRegularizationMethodAManTheoremsabouttheRegulxizationMethodEstimationoftheProbabilityMeasureandProblemofLemingIPrabab~l~lyModelofaRandomExperimentTheBas~cProblemofStahshcsTheBasicProblemsofProbabilityandStatisticsUniformConvergenceofProbabilityMeasureEst~matesCnnrl~i~nnsfortheTTn~formCnmr~rgenreofEstlrnatesintheTTnknnwnPrnhabllltyMeasureStructureofD~stnbut~anFunctionEstlmatarthatProvldesUniformConvergencePmilalUniformConvergenceandGeneralizationofGlivenkoCantelliTheoremDefinitionofPartialUniformConvergenceGeneralizationoftheGlivenkoCantelliProblemMmimizingtheRiskFunctionalUndertheConditionofUniformConvergenceofProbabilityMeasureEstimatesIvnimizingtheRiskFunctionalundertheConditionofPKtialUniformConvergenceofProbabilityMeasureEshmatesRemarksaboutModesofConvergenceoftheProbabilityMeasureEstimatesandStatementsoftheLearnrigProblemConditionsforConsistencyofEmplncalR~skMinimizahonPrincipleClassicalDefinitionofConsistencyDefinitionofStnct(Nontrivial)ConsistencyDefinltlonofStrictConslstencyforthePatternRecagnltionandtheRegress~onEstimationProblemsDefinltlonofStrictConslstencyfortheDensityEstlmat~anProblemEmpiricalProcessesRemarkontheLawofLxgeNumbersandItsGeneralizationTheKeyTheoremofLeamingTheory(TheoremaboutEquivalence)ProofoftheKeyThearemStnctCans~stencyaftheMax~mumL~kel~haodMethodNecessacyandSufficientConditionsforUniformConvergenceofFrequenciestotheirProbabilitiesThreeCasesofUniformConvergenceConditionsofUniformConvergenceintheSimplestModelEntropyofaSetofFunctionsThearemaboutUmformTwoS~dedConvergenceNecessaryandSufficlentCandihonsfarUmformConvergenceofMeanstotheirExpectationsforaSetofRealValuedBoundedFunctionsEntropyofaSetofRealValuedFunctionsTheoremaboutUniformTwoSidedConvergenceNecessaryandSufficlentConditionsfarUmformConvergenceofMeanstothelrExpectationsforSetsofUnboundedFunctionsProofofTheoremKantiProblemofDemarcabanandPopper#sTheoryofNonfalsiflabil~tyITheoremsaboutNonfalsiflabil~tyICaseofCompleteNonfdstfiab~lttyTheoremaboutParbdNonfalstfiabtl~tyThearemaboutPatenhalNanfals$flabtl$tyConQtoniforOneSjdedUrnformConvergenceandConstitencgoftheEmplncdPdskNnlm$rahonPnnapleThreeUlestonesInLeamlngTheoryBaundiantheRtskforIndrcatorLossFunct~oniBoundsfortheStmplestModelPess$mtst$cCaseITheSimplestModelBoundsforthcSimplestModelOpbmtsbcCaseBoundsfortheStmplestModelGeneralCaieTheBastcIneyualtheiPesstmflstflcCaseProofofTheoremITheEastcLemmaProofofBaxcLemmaTheIdeaofProvtngTheoremProofofTheoremBaxcInequallbesGeneralCaiePraafofThearemManNancontmct~reBoundsVCDtmenstonTheStmctureoftheGrowthFunchonCanitructtveDlitrtbutlonFreeBaundionGeneralnabonAb~l~tySolubonofGeneraltredGhuenkoCantell,ProblemPraafafTheoremExampleoftheVCDlmenslanaftheDifferentSetsofFuncboniRemarksabouttheBoundsontheGeneral~zahonAb~lttyofLearntngMach~nesBoundonDevtahonofFrequenctesInTwoHalfSamplesAppendrxtoChapterLowerBoundsantheRlskoftheERMPnnclpleAITwoStrategleimStahsttcalInferenceAMtntmaxLcssStrategyforLeam~ngProblemsAUpperBoundsontheMammalLossfortheEmptrtcalRtskMtn$m~zationPrincipleACptlmlibcCaseAPesslmtsttcCaseALowerBoundfortheMtntmaxLossStrategymtheOpbmtsbcCaseALowerBoundfarMmlmaxLossStrategyfarthePeiilmiihcCaieBoundsantheRtskforRealValuedLaiiFuncbaniBoundsfortheSimplestModelPesstmlsttcCaseConceptsofCapacityFartheSetsofRedVduedFunct~onsNoncanitructweBoundsonGeneral~iabanforSetsofRealValuedFuncbaniTheMatnIdeaConceptsofCapacityFortheSetofRealValuedFunct~onsBoundsfortheGeneralModelPeii$mtihcCaieTheBasicInequaliBIProofofTheoremBoundsfortheGeneralModelUniversalCaseProofofTheoremBoundsforUntFormRelattveConvergenceProofofTheoremfortheCasepProofofTheoremfortheCaseIcpPnorInformat~onforthePL~skMirnm~rauonProbleminSetsofUnboundedLoisFuncboniBoundsontheRiskfarSetsaFUnbaundedNonnegatieFunchonsSampleSelecttonandtheProblemoFOutltersTheMalnResultsoftheTheovofBoundsTheStruchlralRtskMtntmlzahonPnnclpleTheSchemeoftheStructuralRtskMmmmtrattonInducttonPnnctpleIPnnclpleofStructuralRiikMtnimlzabonMtnjrnumDescnpbanLengthandStructuralRiikMtmmtzabanInductivePnnclplesTheIdeaabouttheNatureofRandomPhenomenaStachasbcUlPosedProblemsIStochaittcIIIPosedProblemsRegulanzahonMethodforSolvingStochasticIllPosedProblemsProofsoftheTheoremsProofofTheoremProofofTheoremProofofTheoremConditlanifarCanitstencyoftheMethodsofDenstyEshmationNonparametncEsbmatar~ofDensstyEshmatoriBasedanAppraxmahonsoftheD~stribut~onFunchonbyanEmptncalDlstrtbutlonFunchonTheYarzenEsttmatorsPrqectlonEsttmatorsSpllneEsttmateoftheDensityApproxtmahonbySpl~neioftheOddOrderSpllneEsttmateoftheDensitvApproxtmahonbySpl~neioftheEvenOrderEsttmatorsfortheDtstrlbuttonFunctlonPolygonAppraxmattonofD~stnbuhonFuncaonPrqectlonMethodoftheDensttyEstimatorAsymptoticRateofConvergenceforSmoothDens~vFunctionsProofofTheoremChoosingaVdueofSmoothing(Regulanrabon)ParameterfortheProblemofDens~tyEshmatlonEstlmdtlonoftheRhtloofTwoDensttlesIEsttmahonofCondihonalDens~tieiIEstlmatlanafRattaofTwoDenshesontheLineEst~mhtlonofaCondihondProbabtl~vonaLmeE~tlmnhngtheValuesofFunctlonatCnvenPotntsTheSchemeofMtntmjzjngtheOverallRiskTheMethodofStructuralMlnlmtzatlanoftheOverallR~skBoundsontheUrnformRelativeDeu~htlonofFrequenc~es~nTwoSubsamplesABoundontheUntfarmRelahveDevlatlonofMeansInTwoSubsamplesEshmabanofValuesofanIndrcatarFunctlanInaClassofL~nearDecisionRulesSampleSelect~anfarEstmahngtheValuesofanIndrcatorFuncbanEshmatlonofVduesofaRedFunctlontntheClassofFunctionsLtneartnthemParametersSampleSelecttonforEsttrnahonofValuesofRedValuedFunctlonsLocalAlgonthmiforEstmahngValuesofanIndrcatorFuncbanLocalAlgonthmifarEst~mahngValuesofaRealValuedFunctionTheProblemofFtnngtheBestPointtnaCnvenSetIIChalceoftheMastProbableRepresentattveoftheFsritClaiiChalceoftheBestPamtofalvenSetISupportVectorEstimationofFuuctiourPercepkansandThetrGeneralliat~oniIRosenblauiPerceptranPProofsoftheTheoremsProofofNovtkoffTheoremProofofTheoremMethodofStochashcApproxtmat$anandStgmatdAppraxtmahonofIndrcatorFuncboniMethodofStochasticApproxtmat~onMethodofPatenttalFunctflaniandRadralBassFuncttoniMethodofPotentlaFunct~onstnAsymptottcLearn~ngTheovRaalBasisFunchonMethodThreeTheoremsofOpttmizat~onTheoqFernat#sTheorem()LqrangeMultlpltersRule()KuhnTuckerTheorem()NeuralNetworksTheBackPropqat~onMethodTheBackPropagattonAlgorithmNmalNeWoksFortheRegresslmEst~rnatlonFnoblernRmatkrontheBackqagatlonMethodTheSupportVedmMethodfarEstmahngInd~catmFunctlmsITheOptlmalHypqlaneTheOptimalHypqlmefarNons~parahleSetsTheHadMaqmGenerallrat~onoftheOptimalHyperplaneTheBaslcSolutlonSofiMarglnGenerallrahmStatisticalPropettlesoftheOphmalHyperplaneProofsoftheTheorernshoofofTheomnhoofofTheomnLeaveOneOutProcedureProofofTheomandTheoremProofofTheomProofofTheomnTheIdeaoftheSupportVectorMachlneGenaallratlonInHlghDimensionalSpaceHllbaiSchmtdtTheormdMacerTheorrmConsiructlngSVMachinesOneMoreApproachtotheSuppaltVecbrMethodMnunlzlngtheNwnberofSuppartVedmsGenaallzatlonfortheNmreparableCaseLlnearOptun~zabmMethodforSVMachmesSelectionofSVMachmeUslngBoundsExamplesofSVMachinesforPaitemRecognttlonPolynmlalSuppoitVectorMachmeaRadlalBaelsFunctionSVMach~nerTwo~LayaNeuralSVMach~neeSupportVectorMethodforTransdud~velnfaenceMult~classClass~f~cahmRemarksonGmeral~zat~onoftheSVMethodIITheSuppottVectorMethodforEstunahngRealValuedFunchonsIeInsens~t~veLossFundlonsLossA~ndtonsforRoh~rstEst~matmsMmm~zmgtheRlskwltheInsensltlveLossFunchonsMmirnlzlngtheR~rkforaFlredElementoftheShcturellTheBasicSalut~onsllSolut~onfortheHubaLossFunchonIISVMach~nesforFundlonEstlrnatlonMmunlzlngtheRiskForaFxedElementoftheShctureinFeabrfSpacellTheBaslcSolutionsmFeabreSpaceSolutionforHubaLossFundlonnFeab~reSpacellLlnearOptunlzahonMethodMulhKmelDecampas~t~anaffunchansIConstrudlngKernelsforEshrnat~onofRealValuedFund~onsKernelsGeneratrngExpanslononPolynm~alsConstructingMultidunens~onalKernelsKmelsGeneratingSplinesSpllneofOrdadwlthaFmteNumbaofKnobKanelsGmaatlngSpllneswlthanlnfinlteNumberofKnotsllB,SpllneApproxirnahonsB,Splmesw!fianInfnlleNumberoFKnotsIIKanelsrJenaatlngFourierExpansionsKanelefarRegulanzedFour~aExpms~onsTheSuppaltVedarANOVADecompoa~t~on(SVAD)forFunchmApprox~rni~onandRegress~onEstlrnatlonSVMethodfarSolvingLinearOpemtorEquahoneTheSVMethodIRegulmationbyChoosmgParmeiersofe,InsenslhvityIISVMethodofDens!$EstimationISpllneApproxlrnailonofaDenrltyllApproxlmatlonofaDenrltyw~thGauenanMlxiureIIEsimahonofCond~honalProbab~l~tyandCond~i~onalDene~tyFuncha~sIIIIIEstimahonofCondlhmalhobabllltyFund~msIIIIEstimahonofCond~hmalDens,Fund~onsIConned~onsBetweentheSVMethodandSpaneFunchonApprax~rmt~onllReproducingKemelsH~lbettSpacesModlFledSpaneApproxlrnatlonand~tsRelatlontoSVMachmesSVMachinesforPattanRecopt~onTheQuadraticOpt~rnlzat~onPmblemIIterat~vehocedwefarSpec~fy~ngSuppdVedorsMethodsForSolvlngtheReducedOptm~zat~mhblemDlg~tRecagnlt~onhoblemTheUSPostalSenrlceDatabasePerlmancefortheUSPostalSavlceDatabaseSomeImporiantDetallsComparisonofPafmnceoftheSVMachlnewithGaussianKemeltotheGausslanRBFNetworkTheBestResultsForUSPostalSmrlceDatabaseTangentDistanceDlg~tRecagnlt~onhoblemTheNISTDatabasePerlmanceforNISTDatabaseFuttherImprovementTheBestResultsforNISTDatabaseFubmRaclngOneMoreOppn~$TheTrmsduct~veInfacnceSVMachlnesforFunctlmApproxunahons,RegesslonEstlmatlon,andSlgnalProcessingTheModelSelectionhoblemIFundlmalforModelSelectlonBasedontheVCBoundClarr~calFunchmaleExpamentalCampar~ronofModelSelechmMethodsTheProblemofFeatureSeledlonHasNoGenaalSalubonStrudureontheSetofRegulaniedLinearFund~onsTheLCweMethodTheMethodofEffectweNumberofParmetasTheMethodofEffectiveVCDlmenrlonExpamenisonMeaiunngtheEffedlveVCDlmenr~mFundlmApproxlmailonUalngtheSVMethodWhyDoestheValueofeContmltheNumbaofSupportVectorsSVMachineforRegress~onEshmatlonRoblernofDataSmaothlngEst~mat~onofLmearRegress~onF~rnchonsEshmatlonofNonlmearRegreeelonFund~onsSVMethodforSolvmgthePositronEmlsslmTomogaphy(PET)ProblemDescrlpbonofPETProblemofSolvlngtheRadonEquatlonGenaallrat~onoftheResidualFnncpleofSolvmgPETRoblmsTheClass~calMethodsofSalvlngthePEThblmTheSVMethodforSolvmgthePETRoblmRmakAbouttheSVMethodIllStatirticalFmdatianofLeamhgTheoryNecessaryandSufficlentCondlhonsforUnlfomConvergenceofFt#equenclestothelrPrnbabll~besUnlfmConvsgtllceofkrequenc~estothe~rPmbabd~hesBaelcLemmaEntropyoftheSetofEventsAsymptot~cRopert~esoftheEntropyNecessaryandSufficlentCmdlhmsofUnlfmConvergencePmofofSufflclencyNecessaryandSufficlentCmdlhmsofUnlfmConvergencePmofofNecesslbyNecessaryandSufflclentCondlbmsContlnuabonofProvlngNecesslkjNecessaryandSufficlentCondlhonsforUnlfomConvergenceofMeanstothe~rExpectatlonseEntmpyIIIRoofoftheEx~stenceoftheLmltRoofoftheConvagenceoftheSequtnceTheQuas~mbeeExtens~onofaSetInAulllalyLmaNecessaryandSufflclentCmdlt~oniforTlnlfomCmuagmceTheRoofofpTecessliyNecessaryandSuff~clentCond~honsforUnlfarmConvagmceThehoofofSufficiencyGaollanesErarnThemmLNecessaryandSuff~c~entCondlhonsforTJn~fomOneSldedConvergtnceofMemstoThe~rExpedat~onsInirodud~onMaxmurnVolumeSechmsTheTheoremontheAverageLoganihmThemanontheExlsttnceofaCondmTheoranantheExlsttnceofFund~ansClosetotheComdorBaundar~es(ThemanonPottntlalNonfalsifiab~l~ty)TheNecessaryCondlhonsTheNecessaryandSufficientCmdlbonsConnnentrandBibliogaphicalRemarkrReferencesIndexPagezxIliPREFACEThisbookisdevotedtostatisticallearningtheory,thetheorythatexploreswaysofestimatingfunctionaldependencyfromagivencollectionofdataThisproblemisverygeneralItcoversimportanttopicsofclassicalstatisticsinparticular,discriminantanalysis,regressionanalysis,andthedensityestimationproblemInthisbookweconsideranewparadigmforsolvingtheseproblems:thesocalledlearningparadigmthatwasdevelopedoverthelastyearsIncontrasttotheclassicalstatisticsdevelopedforlargesamplesandbasedonusingvarioustypesofaprioriinformation,thenewtheorywasdevelopedforsmalldatasamplesanddoesnotrelyonaprioriknowledgeaboutaproblemtobesolvedInsteaditconsidersastructureonthesetoffunctionsimplementedbythelearningmachine(asetofnestedsubsetsoffunctions)whereaspecificmeasureofsubsetcapacityisdefinedTocontrolthegeneralizationintheframeworkofthisparadigm,onehastotakeintoaccounttwofactors,namely,thequalityofapproximationofgivendatabythechosenfunctionandthecapacityofthesubsetoffunctionsfromwhichtheapproximatingfunctionwaschosenThisbookpresentsacomprehensivestudyofthistypeofinference(learningprocess)Itcontains:ThegeneralqualitativetheorythatincludesthenecessaryandsufficientconditionsforconsistencyoflearningprocessesThegeneralquantitativetheorythatincludesboundsontherateofconvergence(therateofgeneralization)oftheselearningprocessesPrinciplesforestimatingfunctionsfromasmallcollectionofdatathatarebasedonthedevelopedtheoryMethodsoffunctionestimationandtheirapplicationtosolvingreallifeproblemsthatarebasedontheseprinciplesThebookhasthreeparts:TheoryofLearningandGeneralization,SupportVectorEstimationofFunctions,andStatisticalFoundationofLearningTheoryThefirstpart,TheoryofLearningandGeneralization,analyzesfactorsxxi赵鸣奇高亮赵鸣奇高亮先验信息赵鸣奇高亮赵鸣奇附注由学习机器通过考虑数据的结构来实现赵鸣奇高亮赵鸣奇高亮xxiiPREFACEresponsibleforgeneralizationandshowshowtocontrolthesefactorsinr~rdertogeneralizewellThispartcontainseightchaptersChapterdescribestwodifferentapproachestothelearningproblemThefirstapproachconsiderslearningasaproblemofminimizinganexpectedriskfunctionalinthesituationwhentheprobabilitymeasurethatdefinestheriskisunknownbutiidobservationsaregivenToobtainasolutionintheframeworkofthisapproachonehastosuggestsomeinductiveprincipleThatisonehastodefineaconstructivefunctionalthatshouldbeminimized(insteaduftheexpectedriskfunctional)inordertofindafunctionthatguaranteesasr~~allexpectedlossThesecondapproachconsiderslearningasaproblemofidentificationofthedesiredfunction:Usingobservations,onehastofindthefunctionthatisclosetothedesiredoneIngeneral,thisapproachleadstothenecessityofsolvingthesocalledillposedproblemsChapterdiscussesconnectionsbetweenthemainproblemsoflearningtheoryandproblemsofthefoundationofstatistics,namelytheproblemofestimatingtheprobabilitymeasurefromthedataItdescribestwowaysofestimatingtheprobabilitymeasureOnewayisbasedontheconvergenceofanestimateoftheprobabilitymeasureinaweakmodeandanotherwayisbasedonconvergenceinastrongmodeThesetwowaysofestimatingtheunknownmeasureimplytwoapproachestothelearningproblemdescribedinChapterChapterisdevotedtothequalitativemodeloflearningprocessesnamelytothetheoryofconsistencyofthelearningprocessesbasedontheempiricalriskminimizationinductionprincipleItshowsthatforconsistencyofthelcarningprocessesbasedonthisprincipletheconvergenceofsomeempiricalprocesses(theexistenceofuniformlawoflargenumbers)isnecessaryandsufficientInChaptertheseconditionsarediscussed(Thecorrespondingtheoremswillbeproveninthethirdpartofthebook)ChaptersandestimatetheboundsontherateofconvergenceofthecmpiricalprocessesUsingtheseboundsweobtainboundsontheriskforthefunctionsthatminimizetheempiricalriskfunctionalInChapterweobtainboundsforsetsofindicatorfunctions(forthepatternrecognitionproblem),andinChapterwegeneralizetheseboundsforsetsofrealvaluedfunctions(forregressionestimationproblems)Theboundsdependontwofactors:thevalueofempiricalriskandthecapacityofthesetoffunctionsfromwhichthefunctionminimizingempirica

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