Accelerated variance-reduced methods for saddle-point problems
We consider composite minimax optimization problems where the goal is to find a saddle-point of a large sum of non-bilinear objective functions augmented by simple composite regularizers for the primal and dual variables. For such problems, under the average-smoothness assumption, we propose acceler...
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| Published in: | EURO journal on computational optimization Vol. 10; p. 100048 |
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2022
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| ISSN: | 2192-4406 |
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| Abstract | We consider composite minimax optimization problems where the goal is to find a saddle-point of a large sum of non-bilinear objective functions augmented by simple composite regularizers for the primal and dual variables. For such problems, under the average-smoothness assumption, we propose accelerated stochastic variance-reduced algorithms with optimal up to logarithmic factors complexity bounds. In particular, we consider strongly-convex-strongly-concave, convex-strongly-concave, and convex-concave objectives. To the best of our knowledge, these are the first nearly-optimal algorithms for this setting.
•Optimal accelerated stochastic variance-reduced algorithm for composite saddle-point problems.•Saddle-point problems with different strongly-convex and strongly-concave parameters.•Upper bounds for composite saddle-point problems with a finite sum structure.•Achieving the lower bounds for composite saddle-point problems with finite sum structure up to logarithmic factor. |
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| AbstractList | We consider composite minimax optimization problems where the goal is to find a saddle-point of a large sum of non-bilinear objective functions augmented by simple composite regularizers for the primal and dual variables. For such problems, under the average-smoothness assumption, we propose accelerated stochastic variance-reduced algorithms with optimal up to logarithmic factors complexity bounds. In particular, we consider strongly-convex-strongly-concave, convex-strongly-concave, and convex-concave objectives. To the best of our knowledge, these are the first nearly-optimal algorithms for this setting.
•Optimal accelerated stochastic variance-reduced algorithm for composite saddle-point problems.•Saddle-point problems with different strongly-convex and strongly-concave parameters.•Upper bounds for composite saddle-point problems with a finite sum structure.•Achieving the lower bounds for composite saddle-point problems with finite sum structure up to logarithmic factor. We consider composite minimax optimization problems where the goal is to find a saddle-point of a large sum of non-bilinear objective functions augmented by simple composite regularizers for the primal and dual variables. For such problems, under the average-smoothness assumption, we propose accelerated stochastic variance-reduced algorithms with optimal up to logarithmic factors complexity bounds. In particular, we consider strongly-convex-strongly-concave, convex-strongly-concave, and convex-concave objectives. To the best of our knowledge, these are the first nearly-optimal algorithms for this setting. |
| ArticleNumber | 100048 |
| Author | Kovalev, Dmitry Dvurechensky, Pavel Borodich, Ekaterina Tominin, Vladislav Tominin, Yaroslav Gasnikov, Alexander |
| Author_xml | – sequence: 1 givenname: Ekaterina orcidid: 0000-0003-3339-6550 surname: Borodich fullname: Borodich, Ekaterina email: borodich.ed@phystech.edu organization: Moscow Institute of Physics and Technology, Moscow, Russia – sequence: 2 givenname: Vladislav surname: Tominin fullname: Tominin, Vladislav email: tominin.vd@phystech.edu organization: Moscow Institute of Physics and Technology, Moscow, Russia – sequence: 3 givenname: Yaroslav surname: Tominin fullname: Tominin, Yaroslav email: tominin.yad@phystech.edu organization: Moscow Institute of Physics and Technology, Moscow, Russia – sequence: 4 givenname: Dmitry surname: Kovalev fullname: Kovalev, Dmitry email: dmitry.kovalev@kaust.edu.sa organization: King Abdullah University of Science and Technology, Thuwal, Saudi Arabia – sequence: 5 givenname: Alexander surname: Gasnikov fullname: Gasnikov, Alexander email: gasnikov.av@mipt.ru organization: Moscow Institute of Physics and Technology, Moscow, Russia – sequence: 6 givenname: Pavel orcidid: 0000-0003-1201-2343 surname: Dvurechensky fullname: Dvurechensky, Pavel email: pavel.dvurechensky@wias-berlin.de organization: Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany |
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| Cites_doi | 10.1007/s10957-022-02038-7 10.1111/j.1467-9868.2005.00503.x 10.1080/10556788.2021.1924714 10.1007/s10957-022-02062-7 10.1137/S1052623403422285 10.1007/s10851-010-0251-1 10.2307/1907266 10.1134/S0965542520110020 10.1007/s10107-006-0034-z 10.1134/S096554252101005X 10.1287/moor.2021.1175 10.1007/s10957-015-0771-3 10.1561/2400000003 10.1007/s10107-017-1161-4 10.24033/bsmf.1625 10.1137/S1052623403425629 10.1007/s10107-004-0552-5 |
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| Keywords | Minimax optimization Composite optimization Saddle-point problem Accelerated algorithms Stochastic variance-reduced algorithms |
| Language | English |
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| SubjectTerms | Accelerated algorithms Composite optimization Minimax optimization Saddle-point problem Stochastic variance-reduced algorithms |
| Title | Accelerated variance-reduced methods for saddle-point problems |
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