  In the paper, we study a class of nonconvex nonconcave minimax optimization
problems (i.e., $\min_x\max_y f(x,y)$), where $f(x,y)$ is possible nonconvex in
$x$, and it is nonconcave and satisfies the Polyak-Lojasiewicz (PL) condition
in $y$. Moreover, we propose a class of enhanced momentum-based gradient
descent ascent methods (i.e., MSGDA and AdaMSGDA) to solve these stochastic
Nonconvex-PL minimax problems. In particular, our AdaMSGDA algorithm can use
various adaptive learning rates in updating the variables $x$ and $y$ without
relying on any global and coordinate-wise adaptive learning rates.
Theoretically, we present an effective convergence analysis framework for our
methods. Specifically, we prove that our MSGDA and AdaMSGDA methods have the
best known sample (gradient) complexity of $O(\epsilon^{-3})$ only requiring
one sample at each loop in finding an $\epsilon$-stationary solution (i.e.,
$\mathbb{E}\|\nabla F(x)\|\leq \epsilon$, where $F(x)=\max_y f(x,y)$). This
manuscript commemorates the mathematician Boris Polyak (1935-2023).
