Accelerated Minimax Algorithms Flock Together

Several new accelerated methods in minimax optimization and fixed-point iterations have recently been discovered, and, interestingly, they rely on a mechanism distinct from Nesterov's momentum-based acceleration. In this work, we show that these accelerated algorithms exhibit what we call the merging path (MP) property; the trajectories of these algorithms merge quickly. Using this novel MP property, we establish point convergence of existing accelerated minimax algorithms and derive new state-of-the-art algorithms for the strongly-convex-strongly-concave setup and for the prox-grad setup.