Sliding Methods for H\"older-Smooth Convex--Concave Minimax Optimization with Bilinear Coupling

Abstract

We study convex-concave minimax optimization problems with bilinear coupling of the form $\min_{x\in \mathcal X}\max_{y\in \mathcal Y} \; f(x)+\langle y,\mathbf{B}x\rangle-g(y),$ where the functions $f$ and $g$ have H\"older continuous (sub)gradients. This setting covers a broad range of regimes, from nonsmooth problems with bounded subgradient variation to smooth problems with Lipschitz continuous gradients; for a smooth component used in the coupling-induced regularizer, its Lipschitz-gradient constant is assumed to hold in the ambient space. We propose a sliding method that exploits the composite structure of the problem by querying the oracles associated with $f$, $g$, and the bilinear coupling operator at prescribed frequencies determined by their individual properties. The method is based on a recursive sliding scheme for monotone variational inequalities. We establish convergence guarantees under H\"older continuity and show how the resulting complexity bounds depend explicitly on the H\"older exponents, H\"older constants, strong convexity parameters, and spectral properties of the coupling matrix. Our analysis covers nonstrongly convex and partially strongly convex regimes. For stochastic problems, we prove a uniform expected-gap bound in the degenerate regime and, under ambient smoothness and positive effective curvature, convergence up to an explicit noise floor. Numerical experiments reproduce the predicted H\"older exponents and confirm that the number of gradient evaluations required for each function separates according to its own smoothness level rather than the worse of the two. A tomographic benchmark shows runtime gains when gradient evaluations are more expensive than the additional matrix-vector products.