  Minimax problems are notoriously challenging to optimize. However, we present
that the two-timescale extragradient method can be a viable solution. By
utilizing dynamical systems theory, we show that it converges to points that
satisfy the second-order necessary condition of local minimax points, under
mild conditions that the two-timescale gradient descent ascent fails to work.
This work provably improves upon all previous results on finding local minimax
points, by eliminating a crucial assumption that the Hessian with respect to
the maximization variable is nondegenerate.
