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

arXiv: https://arxiv.org/abs/2104.06206 · HTML: https://mlsanity.com/paper/2104.06206/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Apr 2021.

## Built on by (most cited first, 2 of 2)

- [Convergences for Minimax Optimization Problems over Infinite-Dimensional Spaces Towards Stability in Adversarial Training](https://mlsanity.com/paper/2312.00991/) (2023)
- [Saddle Networks: Structure-Preserving Architectures for Convex-Concave Functions](https://mlsanity.com/paper/2605.28894/) (2026)

## Abstract (arXiv)

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(frac1K)  and linear convergence like O(θK) with  θ < 1 , respectively. In terms of function values we obtain ergodic convergence rates of order  O(frac1K) ,  O(frac1K2)  and  O(θK)  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.