  Unlike nonconvex optimization, where gradient descent is guaranteed to
converge to a local optimizer, algorithms for nonconvex-nonconcave minimax
optimization can have topologically different solution paths: sometimes
converging to a solution, sometimes never converging and instead following a
limit cycle, and sometimes diverging. In this paper, we study the limiting
behaviors of three classic minimax algorithms: gradient descent ascent (GDA),
alternating gradient descent ascent (AGDA), and the extragradient method (EGM).
Numerically, we observe that all of these limiting behaviors can arise in
Generative Adversarial Networks (GAN) training and are easily demonstrated for
a range of GAN problems. To explain these different behaviors, we study the
high-order resolution continuous-time dynamics that correspond to each
algorithm, which results in the sufficient (and almost necessary) conditions
for the local convergence by each method. Moreover, this ODE perspective allows
us to characterize the phase transition between these different limiting
behaviors caused by introducing regularization as Hopf Bifurcations.
