  Despite the established convergence theory of Optimistic Gradient Descent
Ascent (OGDA) and Extragradient (EG) methods for the convex-concave minimax
problems, little is known about the theoretical guarantees of these methods in
nonconvex settings. To bridge this gap, for the first time, this paper
establishes the convergence of OGDA and EG methods under the
nonconvex-strongly-concave (NC-SC) and nonconvex-concave (NC-C) settings by
providing a unified analysis through the lens of single-call extra-gradient
methods. We further establish lower bounds on the convergence of GDA/OGDA/EG,
shedding light on the tightness of our analysis. We also conduct experiments
supporting our theoretical results. We believe our results will advance the
theoretical understanding of OGDA and EG methods for solving complicated
nonconvex minimax real-world problems, e.g., Generative Adversarial Networks
(GANs) or robust neural networks training.
