An elementary approach to tight worst case complexity analysis of gradient based methods
This work presents a novel analysis that allows to achieve tight complexity bounds of gradient-based methods for convex optimization. We start by identifying some of the pitfalls rooted in the classical complexity analysis of the gradient descent method, and show how they can be remedied. Our method...
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| Veröffentlicht in: | Mathematical programming Jg. 201; H. 1-2; S. 63 - 96 |
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| Format: | Journal Article |
| Sprache: | Englisch |
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Springer Berlin Heidelberg
01.09.2023
Springer |
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| ISSN: | 0025-5610, 1436-4646 |
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| Abstract | This work presents a novel analysis that allows to achieve tight complexity bounds of gradient-based methods for convex optimization. We start by identifying some of the pitfalls rooted in the classical complexity analysis of the gradient descent method, and show how they can be remedied. Our methodology hinges on elementary and direct arguments in the spirit of the classical analysis. It allows us to establish some new (and reproduce known) tight complexity results for several fundamental algorithms including, gradient descent, proximal point and proximal gradient methods which previously could be proven only through computer-assisted convergence proof arguments. |
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| AbstractList | This work presents a novel analysis that allows to achieve tight complexity bounds of gradient-based methods for convex optimization. We start by identifying some of the pitfalls rooted in the classical complexity analysis of the gradient descent method, and show how they can be remedied. Our methodology hinges on elementary and direct arguments in the spirit of the classical analysis. It allows us to establish some new (and reproduce known) tight complexity results for several fundamental algorithms including, gradient descent, proximal point and proximal gradient methods which previously could be proven only through computer-assisted convergence proof arguments. |
| Audience | Academic |
| Author | Teboulle, Marc Vaisbourd, Yakov |
| Author_xml | – sequence: 1 givenname: Marc orcidid: 0000-0002-4228-131X surname: Teboulle fullname: Teboulle, Marc email: teboulle@tauex.tau.ac.il organization: School of Mathematical Sciences, Tel-Aviv University – sequence: 2 givenname: Yakov surname: Vaisbourd fullname: Vaisbourd, Yakov organization: Department of Mathematics and Statistics, McGill University |
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| Cites_doi | 10.24033/bsmf.1625 10.1137/0716071 10.1137/080716542 10.1007/s10107-016-1009-3 10.1016/0022-247X(79)90234-8 10.1016/0041-5553(66)90114-5 10.1007/s10107-018-1284-2 10.1090/S0002-9904-1964-11178-2 10.1137/16M108104X 10.1137/1.9781611974997 10.1137/0329022 10.1007/s10957-020-01770-2 10.1007/s10107-013-0653-0 |
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| Copyright | Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2022. Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. COPYRIGHT 2023 Springer |
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| Keywords | Convex minimization Gradient descent 68Q25 Performance estimation problem Proximal schemes 90C30 Composite minimization Global rate of convergence 90C25 90C60 Worst-case complexity analysis |
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| References | CR4 Bertsekas (CR3) 2015 Goldstein (CR6) 1964; 70 Graham, Knuth, Patashnik (CR8) 1994 Taylor, Hendrickx, Glineur (CR16) 2017; 161 Beck, Teboulle (CR1) 2009; 2 Levitin, Polyak (CR10) 1966; 6 Lions, Mercier (CR11) 1979; 16 Moreau (CR13) 1965; 93 Teboulle (CR18) 2018; 170 Beck (CR2) 2017 Passty (CR14) 1979; 72 Drori, Teboulle (CR5) 2014; 145 Güler (CR7) 1991; 29 Sabach, Teboulle (CR15) 2019; 20 Taylor, Hendrickx, Glineur (CR17) 2017; 27 Kim, Fessler (CR9) 2021; 188 Martinet (CR12) 1970; 4 O Güler (1899_CR7) 1991; 29 R Graham (1899_CR8) 1994 J-J Moreau (1899_CR13) 1965; 93 A Beck (1899_CR2) 2017 A Beck (1899_CR1) 2009; 2 1899_CR4 B Martinet (1899_CR12) 1970; 4 DP Bertsekas (1899_CR3) 2015 D Kim (1899_CR9) 2021; 188 ES Levitin (1899_CR10) 1966; 6 GB Passty (1899_CR14) 1979; 72 M Teboulle (1899_CR18) 2018; 170 AA Goldstein (1899_CR6) 1964; 70 PL Lions (1899_CR11) 1979; 16 S Sabach (1899_CR15) 2019; 20 Y Drori (1899_CR5) 2014; 145 AB Taylor (1899_CR17) 2017; 27 AB Taylor (1899_CR16) 2017; 161 |
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| Title | An elementary approach to tight worst case complexity analysis of gradient based methods |
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