Nonsmooth Nonconvex-Concave Minimax Optimization: Convergence Criteria and Algorithms

TL;DR

This paper introduces new convergence criteria and algorithms for constrained stochastic nonsmooth minimax problems, focusing on nonconvex-nonsmooth settings with theoretical guarantees.

Contribution

It develops projected gradient-free descent ascent methods with non-asymptotic convergence analysis for nonsmooth, nonconvex minimax problems, without requiring weak convexity assumptions.

Findings

Defined the notion of Goldstein saddle stationary points for convergence characterization.

Proposed algorithms achieve non-asymptotic convergence rates for finding stationary points.

The methods do not rely on weak convexity assumptions used in prior works.

Abstract

This paper considers constrained stochastic nonsmooth minimax optimization problem of the form $\min_{\mathbf{x}\in\mathcal{X}}\max_{\mathbf{y}\in\mathcal{Y}}f\left(\mathbf{x},\mathbf{y}\right)=\mathbb{E}[F(\mathbf{x},\mathbf{y};\mathbf{\xi})]$, where the objective $f(\mathbf{x},\mathbf{y})$ is concave in $\mathbf{y}$ but possibly nonconvex in $\mathbf{x}$, the stochastic component $F(\mathbf{x},\mathbf{y};\mathbf{\xi})$ indexed by random variable $\mathbf{\xi}$ is mean-squared Lipschitz continuous, and the feasible sets $\mathcal X$ and $\mathcal Y$ are convex and compact. We introduce the notion of $(\eta_x,\eta_y,\delta,\epsilon)$-Goldstein saddle stationary point (GSSP) to characterize the convergence for solving constrained nonsmooth minimax problems. We then develop projected gradient-free descent ascent methods for finding $(\eta_x,\eta_y,\delta,\epsilon)$-GSSPs of the objective function $f(\mathbf{x},\mathbf{y})$ with non-asymptotic convergence rates. We further propose nested-loop projected gradient-free descent ascent methods to establish the non-asymptotic convergence for finding $(\eta,\delta,\epsilon)$-generalized Goldstein stationary points (GGSP) [Liu et al., 2024] of the primal function $\Phi(\mathbf{x})\triangleq\max_{\mathbf{y}\in\mathcal{Y}}{f}\left(\mathbf{x},\mathbf{y}\right)$. It is worth noting that our algorithm designs and theoretical analyses do not require additional assumptions such as the weak convexity used in prior works on nonsmooth minimax optimization [Lin et al., 2025, Bo\c{t} and B\"ohm, 2023].