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.

翻译：摘要：近年来，极小极大优化问题因其在机器学习中的广泛应用而受到越来越多的关注。然而，现有的大多数极小极大求解技术要么是单机算法，要么是由中心服务器协调的分布式算法。本文聚焦于面向带域约束学习的去中心化极小极大优化问题，其中多个智能体在无需任何服务器协调的情况下，共同求解一个非凸-强凹的极小极大鞍点问题。带域约束的去中心化极小极大优化问题支撑着诸多重要的机器学习应用，包括多智能体机器学习公平性保障以及多智能体强化学习中的策略评估。我们提出了一种名为PRECISION（近端梯度跟踪与随机递归方差缩减）的算法，其收敛速率为$O(1/T)$，其中$T$为最大迭代次数。为进一步降低样本复杂度，我们提出了采用自适应批量大小技术的PRECISION$^+$算法。我们证明，PRECISION和PRECISION$^+$到$\epsilon$-稳定点的快速$O(1/T)$收敛性意味着$O(\epsilon^{-2})$的通信复杂度和$O(m\sqrt{n}\epsilon^{-2})$的样本复杂度，其中$m$为智能体数量，$n$为每个智能体上的数据集大小。据我们所知，这是首个在带域约束的去中心化极小极大学习中同时实现$O(\epsilon^{-2})$样本与通信复杂度的工作。我们的实验也验证了理论结果。