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 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 $O\left(\frac{1}{K}\right)$ and linear convergence like $O\left({\theta }^K\right)$ with $\theta <1$, respectively. In terms of function values we obtain ergodic convergence rates of order $O\left(\frac{1}{K}\right)$, $O\left(\frac{1}{K^2}\right)$ and $O\left({\theta }^K\right)$ with $\theta <1$, 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.