  The gradient descent-ascent (GDA) algorithm has been widely applied to solve
minimax optimization problems. In order to achieve convergent policy parameters
for minimax optimization, it is important that GDA generates convergent
variable sequences rather than convergent sequences of function values or
gradient norms. However, the variable convergence of GDA has been proved only
under convexity geometries, and there lacks understanding for general nonconvex
minimax optimization. This paper fills such a gap by studying the convergence
of a more general proximal-GDA for regularized nonconvex-strongly-concave
minimax optimization. Specifically, we show that proximal-GDA admits a novel
Lyapunov function, which monotonically decreases in the minimax optimization
process and drives the variable sequence to a critical point. By leveraging
this Lyapunov function and the K{\L} geometry that parameterizes the local
geometries of general nonconvex functions, we formally establish the variable
convergence of proximal-GDA to a critical point $x^*$, i.e., $x_t\to x^*,
y_t\to y^*(x^*)$. Furthermore, over the full spectrum of the
K{\L}-parameterized geometry, we show that proximal-GDA achieves different
types of convergence rates ranging from sublinear convergence up to finite-step
convergence, depending on the geometry associated with the K{\L} parameter.
This is the first theoretical result on the variable convergence for nonconvex
minimax optimization.
