  Minimax optimization has found extensive applications in modern machine
learning, in settings such as generative adversarial networks (GANs),
adversarial training and multi-agent reinforcement learning. As most of these
applications involve continuous nonconvex-nonconcave formulations, a very basic
question arises---"what is a proper definition of local optima?"
  Most previous work answers this question using classical notions of
equilibria from simultaneous games, where the min-player and the max-player act
simultaneously. In contrast, most applications in machine learning, including
GANs and adversarial training, correspond to sequential games, where the order
of which player acts first is crucial (since minimax is in general not equal to
maximin due to the nonconvex-nonconcave nature of the problems). The main
contribution of this paper is to propose a proper mathematical definition of
local optimality for this sequential setting---local minimax, as well as to
present its properties and existence results. Finally, we establish a strong
connection to a basic local search algorithm---gradient descent ascent (GDA):
under mild conditions, all stable limit points of GDA are exactly local minimax
points up to some degenerate points.
