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Recently, min-max optimization problems have received increasing attentiondue to their wide range of applications in machine learning (ML). However, mostexisting min-max solution techniques are either single-machine or distributedalgorithms coordinated by a central server. In this paper, we focus on thedecentralized min-max optimization for learning with domain constraints, wheremultiple agents collectively solve a nonconvex-strongly-concave min-max saddlepoint problem without coordination from any server. Decentralized min-maxoptimization problems with domain constraints underpins many important MLapplications, including multi-agent ML fairness assurance, and policyevaluations in multi-agent reinforcement learning. We propose an algorithmcalled PRECISION (proximal gradient-tracking and stochastic recursive variancereduction) that enjoys a convergence rate of $O(1/T)$, where $T$ is the maximumnumber of iterations. To further reduce sample complexity, we proposePRECISION$^+$ with an adaptive batch size technique. We show that the fast$O(1/T)$ convergence of PRECISION and PRECISION$^+$ to an $\epsilon$-stationarypoint imply $O(\epsilon^{-2})$ communication complexity and$O(m\sqrt{n}\epsilon^{-2})$ sample complexity, where $m$ is the number ofagents and $n$ is the size of dataset at each agent. To our knowledge, this isthe first work that achieves $O(\epsilon^{-2})$ in both sample andcommunication complexities in decentralized min-max learning with domainconstraints. Our experiments also corroborate the theoretical results.

PRECISION: Decentralized Constrained Min-Max Learning with Low Communication and Sample Complexities