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1、TheoryofConvexOptimizationforMachineLearningSebastienBubeck11DepartmentofOperationsResearchandFinancialEngineering,PrincetonUniversity,Princeton08544,USA,sbubeck@princeton.eduAbstractThismonographpresentsthemainmathematicalideasinconvexopti-mization.
2、Startingfromthefundamentaltheoryofblack-boxoptimiza-tion,thematerialprogressestowardsrecentadvancesinstructuralop-timizationandstochasticoptimization.Ourpresentationofblack-boxoptimization,stronglyin
uencedbytheseminalbookofNesterov,in-cludestheanalys
3、isoftheEllipsoidMethod,aswellas(accelerated)gra-dientdescentschemes.Wealsopayspecialattentiontonon-Euclideansettings(relevantalgorithmsincludeFrank-Wolfe,MirrorDescent,andDualAveraging)anddiscusstheirrelevanceinmachinelearning.Weprovideagentleintroduc
4、tiontostructuraloptimizationwithFISTA(tooptimizeasumofasmoothandasimplenon-smoothterm),Saddle-PointMirrorProx(Nemirovski'salternativetoNesterov'ssmoothing),andaconcisedescriptionofInteriorPointMethods.Instochasticop-timizationwediscussStochasticGradie
5、ntDescent,mini-batches,Ran-domCoordinateDescent,andsublinearalgorithms.Wealsobrie
ytouchuponconvexrelaxationofcombinatorialproblemsandtheuseofrandomnesstoroundsolutions,aswellasrandomwalksbasedmethods.Contents1Introduction11.1Someconvexoptimizationpro
6、blemsformachinelearning21.2Basicpropertiesofconvexity31.3Whyconvexity?61.4Black-boxmodel71.5Structuredoptimization81.6Overviewoftheresults92Convexoptimizationin nitedimension122.1Thecenterofgravitymethod122.2Theellipsoidmethod143Dimension-freeconvexop
7、timization193.1ProjectedSubgradientDescentforLipschitzfunctions203.2Gradientdescentforsmoothfunctions233.3ConditionalGradientDescent,akaFrank-Wolfe283.4Strongconvexity33iiiContents3.5Lowerbounds373.6Nesterov'sAcceleratedGradientDescent414Almostdimensi
8、on-freeconvexoptimizationinnon-Euclideanspaces484.1Mirrormaps504.2MirrorDescent514.3StandardsetupsforMirrorDescent534.4LazyMirrorDescent,akaNesterov'sDualAveraging554.5MirrorProx574.6Thevector eldpointofviewonMD,DA,andMP595Beyondtheblack-boxmo
