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Convex Optimization: Algorithms and Complexity

Bubeck, Sébastien·2015·Foundations and Trends® in Machine Learning
DOI 10.1561/2200000050
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Abstract

This monograph presents the main complexity theorems in convex optimization and their corresponding algorithms. Starting from the fundamental theory of black-box optimization, the material progresses towards recent advances in structural optimization and stochastic optimization. Our presentation of black-box optimization, strongly influenced by Nesterov’s seminal book and Nemirovski’s lecture notes, includes the analysis of cutting plane methods, as well as (accelerated) gradient descent schemes. We also pay special attention to non-Euclidean settings (relevant algorithms include Frank-Wolfe, mirror descent, and dual averaging) and discuss their relevance in machine learning. We provide a gentle introduction to structural optimization with FISTA (to optimize a sum of a smooth and a simple non-smooth term), saddle-point mirror prox (Nemirovski’s alternative to Nesterov’s smoothing), and a concise description of interior point methods. In stochastic optimization we discuss stochastic gradient descent, minibatches, random coordinate descent, and sublinear algorithms. We also briefly touch upon convex relaxation of combinatorial problems and the use of randomness to round solutions, as well as random walks based methods.

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Metadata

Authors
Bubeck, S.
Affiliations
Theory Group, Microsoft Research
Venue
Foundations and Trends® in Machine Learning
Publisher
Emerald
Published
2015-11-12
Volume
8
Issue
3-4
Pages
231-357
ISSN
  • 1935-8237
  • 1935-8245
DOI
10.1561/2200000050
Cited by
1,001
References
7
Language
en

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Bubeck, S. (2015). Convex Optimization: Algorithms and Complexity. Foundations and Trends® in Machine Learning. https://doi.org/10.1561/2200000050

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