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ConvexOptimization:Algorithmsand Complexity

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 ..2583 Dimension-free convex Projected subgradient descent for Lipschitz functions .. Gradient descent for smooth functions .. Conditional gradient descent, aka Frank-Wolfe .. Strong convexity .. Lower bounds .. Geometric descent .. Nesterov s accelerated gradient descent ..2894 Almost dimension-free convex optimization in Mirror maps .. Mirror descent .. Standard setups for mirror descent .. Lazy mirror descent, aka Nesterov s dual averaging.

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-Euclidean settings (relevant algorithms include Frank-Wolfe, mirror

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