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ConvexOptimization:Algorithmsand Complexity

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Foundations and TrendsR in Machine LearningVol. 8, No. 3-4 (2015) 231 357c 2015 S. BubeckDOI: optimization : Algorithms andComplexityS bastien BubeckTheory Group, Microsoft Some convex optimization problems in machine learning . Basic properties of convexity . . . . . . . . . . . . . . . . Why convexity? . . . . . . . . . . . . . . . . . . . . . . . Black-box model . . . . . . . . . . . . . . . . . . . . . . . Structured optimization . . . . . . . . . . . . . . . . . . . Overview of the results and disclaimer . . . . . . . . . . .2402 Convex optimization in finite The center of gravity method . . . . . . . . . . . . . . . . The ellipsoid method . . . . . . . . . . . . . . . . . . . . Vaidya s cutting plane method . . . . . . . . . . . . . . . Conjugate gradient.

wards recent advances in structural optimization and stochastic op-timization. Our presentation of black-box optimization, strongly in-fluenced by Nesterov’s seminal book and Nemirovski’s lecture notes, includes the analysis of cutting plane methods, as well as (acceler-ated)gradientdescentschemes.Wealsopayspecialattentiontonon-

  Optimization, Optimiza tion, Timization

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