A variance reduced framework for (non)smooth nonconvex-nonconcave stochastic minimax problems with extended Kurdyka-Lojasiewicz property

TL;DR

This paper introduces a variance reduced algorithm for complex stochastic minimax problems with weak convexity and the extended Kurdyka-Lojasiewicz property, broadening applicability and achieving optimal sample complexity.

Contribution

It presents the first unified framework handling weak convexity, extended KL property, and variance reduction in stochastic minimax optimization, with state-of-the-art convergence guarantees.

Findings

Achieves optimal sample complexity in smooth finite-sum setting.

Handles a broad class of nonconvex-nonconcave problems.

Provides convergence guarantees under minimal assumptions.

Abstract

In this paper, we study stochastic constrained minimax optimization problems with nonconvex-nonconcave structure, a central problem in modern machine learning, for which reliable and efficient algorithms remain largely unexplored due to its inherent challenges. Prior approaches for nonconvex minimax optimization often require (strong) concavity on the maximization part, or certain restrictive geometric assumptions on the joint objective to have guaranteed convergence. In contrast, our method only assumes weak convexity in the primal variable and the extended Kurdyka-Lojasiewicz (KL) property, with exponent $\theta \in [0,1]$, in the dual variable, significantly broadening the class of tractable problems. To this end, we propose a variance reduced algorithm that provably handles this general setting and achieves an $\varepsilon$-stationary solution with state-of-the-art sample complexity: in the smooth finite-sum setting, the sample complexity is $\mathcal{O}\left(\sqrt{N}\,\varepsilon^{-\max\{4\theta,2\}}\right)$, where $N$ is the number of total samples, and in the online smooth setting, it is $\mathcal{O}\Big(\varepsilon^{-\max\{6\theta,3\}}\Big)$. For the structured nonsmooth problem, the sample complexity is $\mathcal{O}\left(\sqrt{N}\,\max\Big\{\varepsilon^{-3}, \varepsilon^{-5\theta}, \varepsilon^{-\frac{11\theta-3}{2\theta}}\Big\}\right)$ and $\mathcal{O}\left(\max\left\{\varepsilon^{-4}, \varepsilon^{-\frac{15\theta-1}{2}}, \varepsilon^{-\frac{31\theta-9}{4\theta}}\right\}\right)$ respectively for the two settings. To the best of our knowledge, this is the first unified framework that jointly accommodates weak convexity, the extended KL property, and variance-reduced stochastic updates, making it highly suitable for large-scale applications.