Weaker MVI Condition: Extragradient Methods with Multi-Step Exploration

This paper proposes a new framework of algorithms that is extended from the celebrated extragradient algorithm. The min-max problem has attracted increasing attention because of its applications in machine learning tasks such as generative adversarial networks (GANs) training. While there has been exhaustive research on convex-concave setting, problem of nonconvex-nonconcave setting faces many challenges, such as convergence to limit cycles. Given that general min-max optimization has been found to be intractable, recent research efforts have shifted towards tackling structured problems. One of these follows the weak Minty variational inequality (weak MVI), which is motivated by relaxing Minty variational inequality (MVI) without compromising convergence guarantee of extragradient algorithm. Existing extragradient-type algorithms involve one exploration step and one update step per iteration. We analyze the algorithms with multiple exploration steps and show that current assumption can be further relaxed when more exploration is introduced. Furthermore, we design an adaptive algorithm that explores until the optimal improvement is achieved. This process exploits information from the whole trajectory and effectively tackles cyclic behaviors.