A Unified Zeroth-Order Approach for Decentralized Minimax Optimization

Abstract

We propose ZOMA, a unified Zeroth-Order decentralized accelerated MinimAx framework for multi-agent nonconvex Polyak--Łojasiewicz minimax optimization. The proposed framework only requires evaluating the function value and, as such, is tailored to gradient-free environments, where exact gradient information is either unavailable or computationally prohibitive to obtain. A central contribution of our \textbf{ZOMA} framework is a multi-level unification, along the following directions: (i) \emph{estimator} - our framework adopts a hybrid zeroth-order estimator, which accommodates, among others, both coordinate-wise and randomized uniform smoothing estimators; (ii) \emph{bias correction} - our framework subsumes a wide range of bias-correction strategies, including gradient tracking (GT), exact diffusion (ED), and EXTRA and (iii) \emph{acceleration} - our framework facilitates a broad class of acceleration techniques, including zeroth-order versions of STORM, PAGE, and L2S. The general nature of \textbf{ZOMA} leads to many novel decentralized zeroth-order minimax methods and allows us to establish unified convergence guarantees, matching the performance of state-of-the-art centralized zeroth-order minimax methods, while providing benefits, such as linear speed-up in the number of users. The unified framework also provides a systematic way to assess algorithmic suitability by specializing the convergence rates to specific problem structures and method designs. We validate the performance of the proposed algorithms via numerical simulations.