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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Bibliographic Details
Published in:Computational optimization and applications Vol. 86; no. 3; pp. 925 - 966
Main Authors: Boţ, Radu Ioan, Csetnek, Ernö Robert, Sedlmayer, Michael
Format: Journal Article
Language:English
Published: New York Springer US 01.12.2023
Springer Nature B.V
Subjects:
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
Online Access:Get full text
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Summary: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.
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ISSN:0926-6003
1573-2894
DOI:10.1007/s10589-022-00378-8