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An accelerated minimax algorithm for convex-concave saddle point problems with nonsmooth coupling function

In this work we aim to solve a convex-concave saddle point problem, where the convex-concave coupling function is smooth in one variable and nonsmooth in the other and not assumed to be linear in either. The problem is augmented by a nonsmooth regulariser in the smooth component. We propose and inve...

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Autores principales: Boţ, Radu Ioan, Csetnek, Ernö Robert, Sedlmayer, Michael
Formato: Online Artículo Texto
Lenguaje:English
Publicado: Springer US 2022
Materias:
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10643324/
https://www.ncbi.nlm.nih.gov/pubmed/37969869
http://dx.doi.org/10.1007/s10589-022-00378-8
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author Boţ, Radu Ioan
Csetnek, Ernö Robert
Sedlmayer, Michael
author_facet Boţ, Radu Ioan
Csetnek, Ernö Robert
Sedlmayer, Michael
author_sort Boţ, Radu Ioan
collection PubMed
description In this work we aim to solve a convex-concave saddle point problem, where the convex-concave coupling function is smooth in one variable and nonsmooth in the other and not assumed to be linear in either. The problem is augmented by a nonsmooth regulariser in the smooth component. We propose and investigate a novel algorithm under the name of OGAProx, consisting of an optimistic gradient ascent step in the smooth variable coupled with a proximal step of the regulariser, and which is alternated with a proximal step in the nonsmooth component of the coupling function. We consider the situations convex-concave, convex-strongly concave and strongly convex-strongly concave related to the saddle point problem under investigation. Regarding iterates we obtain (weak) convergence, a convergence rate of order [Formula: see text] and linear convergence like [Formula: see text] with [Formula: see text] , respectively. In terms of function values we obtain ergodic convergence rates of order [Formula: see text] , [Formula: see text] and [Formula: see text] with [Formula: see text] , respectively. We validate our theoretical considerations on a nonsmooth-linear saddle point problem, the training of multi kernel support vector machines and a classification problem incorporating minimax group fairness.
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spelling pubmed-106433242023-11-14 An accelerated minimax algorithm for convex-concave saddle point problems with nonsmooth coupling function Boţ, Radu Ioan Csetnek, Ernö Robert Sedlmayer, Michael Comput Optim Appl Article In this work we aim to solve a convex-concave saddle point problem, where the convex-concave coupling function is smooth in one variable and nonsmooth in the other and not assumed to be linear in either. The problem is augmented by a nonsmooth regulariser in the smooth component. We propose and investigate a novel algorithm under the name of OGAProx, consisting of an optimistic gradient ascent step in the smooth variable coupled with a proximal step of the regulariser, and which is alternated with a proximal step in the nonsmooth component of the coupling function. We consider the situations convex-concave, convex-strongly concave and strongly convex-strongly concave related to the saddle point problem under investigation. Regarding iterates we obtain (weak) convergence, a convergence rate of order [Formula: see text] and linear convergence like [Formula: see text] with [Formula: see text] , respectively. In terms of function values we obtain ergodic convergence rates of order [Formula: see text] , [Formula: see text] and [Formula: see text] with [Formula: see text] , respectively. We validate our theoretical considerations on a nonsmooth-linear saddle point problem, the training of multi kernel support vector machines and a classification problem incorporating minimax group fairness. Springer US 2022-06-04 2023 /pmc/articles/PMC10643324/ /pubmed/37969869 http://dx.doi.org/10.1007/s10589-022-00378-8 Text en © The Author(s) 2022, corrected publication 2022 https://creativecommons.org/licenses/by/4.0/Open AccessThis article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/ (https://creativecommons.org/licenses/by/4.0/) .
spellingShingle Article
Boţ, Radu Ioan
Csetnek, Ernö Robert
Sedlmayer, Michael
An accelerated minimax algorithm for convex-concave saddle point problems with nonsmooth coupling function
title An accelerated minimax algorithm for convex-concave saddle point problems with nonsmooth coupling function
title_full An accelerated minimax algorithm for convex-concave saddle point problems with nonsmooth coupling function
title_fullStr An accelerated minimax algorithm for convex-concave saddle point problems with nonsmooth coupling function
title_full_unstemmed An accelerated minimax algorithm for convex-concave saddle point problems with nonsmooth coupling function
title_short An accelerated minimax algorithm for convex-concave saddle point problems with nonsmooth coupling function
title_sort accelerated minimax algorithm for convex-concave saddle point problems with nonsmooth coupling function
topic Article
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10643324/
https://www.ncbi.nlm.nih.gov/pubmed/37969869
http://dx.doi.org/10.1007/s10589-022-00378-8
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