Decentralized minimax optimization has been actively studied in the past few years due to its application in a wide range of machine learning models. However, the current theoretical understanding of its convergence rate is far from satisfactory since existing works only focus on the nonconvex-strongly-concave problem. This motivates us to study decentralized minimax optimization algorithms for the nonconvex-nonconcave problem. To this end, we develop two novel decentralized stochastic variance-reduced gradient descent ascent algorithms for the finite-sum nonconvex-nonconcave problem that satisfies the Polyak-{\L}ojasiewicz (PL) condition. In particular, our theoretical analyses demonstrate how to conduct local updates and perform communication to achieve the linear convergence rate. To the best of our knowledge, this is the first work achieving linear convergence rates for decentralized nonconvex-nonconcave problems. Finally, we verify the performance of our algorithms on both synthetic and real-world datasets. The experimental results confirm the efficacy of our algorithms.

翻译：去中心化极小极大优化因广泛适用于各类机器学习模型，近年来备受关注。然而，现有研究仅聚焦于非凸-强凹问题，其收敛速率的理论理解仍远未令人满意。这促使我们针对非凸-非凹问题探索去中心化极小极大优化算法。为此，我们针对满足Polyak-Łojasiewicz（PL）条件的有限和式非凸-非凹问题，开发了两种新颖的去中心化随机方差缩减梯度上升-下降算法。理论分析揭示了如何通过局部更新与通信机制实现线性收敛速率。据我们所知，这是首个实现去中心化非凸-非凹问题线性收敛速率的研究。最后，我们在合成数据集与真实数据集上验证了算法性能，实验结果证实了算法的有效性。