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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Vydáno v:	Computational optimization and applications Ročník 86; číslo 3; s. 925 - 966
Hlavní autoři:	Boţ, Radu Ioan, Csetnek, Ernö Robert, Sedlmayer, Michael
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	New York Springer US 01.12.2023 Springer Nature B.V
Témata:	Algorithms Convergence Convex and Discrete Geometry Coupling Investigations Management Science Mathematics Mathematics and Statistics Minimax technique Operations Research Operations Research/Decision Theory Optimization Saddle points Statistics Support vector machines Convergence rate Linear convergence Saddle point problem Minimax algorithm Convex-concave Acceleration
ISSN:	0926-6003, 1573-2894
On-line přístup:	Získat plný text
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Popis
Shrnutí:	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 O ( 1 K ) and linear convergence like O ( θ K ) with θ < 1 , respectively. In terms of function values we obtain ergodic convergence rates of order O ( 1 K ) , O ( 1 K 2 ) and O ( θ K ) with θ < 1 , 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.
Bibliografie:	ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 content type line 23
ISSN:	0926-6003 1573-2894
DOI:	10.1007/s10589-022-00378-8