  Recently, min-max optimization problems have received increasing attention
due to their wide range of applications in machine learning (ML). However, most
existing min-max solution techniques are either single-machine or distributed
algorithms coordinated by a central server. In this paper, we focus on the
decentralized min-max optimization for learning with domain constraints, where
multiple agents collectively solve a nonconvex-strongly-concave min-max saddle
point problem without coordination from any server. Decentralized min-max
optimization problems with domain constraints underpins many important ML
applications, including multi-agent ML fairness assurance, and policy
evaluations in multi-agent reinforcement learning. We propose an algorithm
called PRECISION (proximal gradient-tracking and stochastic recursive variance
reduction) that enjoys a convergence rate of $O(1/T)$, where $T$ is the maximum
number of iterations. To further reduce sample complexity, we propose
PRECISION$^+$ with an adaptive batch size technique. We show that the fast
$O(1/T)$ convergence of PRECISION and PRECISION$^+$ to an $\epsilon$-stationary
point imply $O(\epsilon^{-2})$ communication complexity and
$O(m\sqrt{n}\epsilon^{-2})$ sample complexity, where $m$ is the number of
agents and $n$ is the size of dataset at each agent. To our knowledge, this is
the first work that achieves $O(\epsilon^{-2})$ in both sample and
communication complexities in decentralized min-max learning with domain
constraints. Our experiments also corroborate the theoretical results.
