  Nonconvex-nonconcave minimax optimization has received intense attention over
the last decade due to its broad applications in machine learning. Most
existing algorithms rely on one-sided information, such as the convexity (resp.
concavity) of the primal (resp. dual) functions, or other specific structures,
such as the Polyak-\L{}ojasiewicz (P\L{}) and Kurdyka-\L{}ojasiewicz (K\L{})
conditions. However, verifying these regularity conditions is challenging in
practice. To meet this challenge, we propose a novel universally applicable
single-loop algorithm, the doubly smoothed gradient descent ascent method
(DS-GDA), which naturally balances the primal and dual updates. That is, DS-GDA
with the same hyperparameters is able to uniformly solve nonconvex-concave,
convex-nonconcave, and nonconvex-nonconcave problems with one-sided K\L{}
properties, achieving convergence with $\mathcal{O}(\epsilon^{-4})$ complexity.
Sharper (even optimal) iteration complexity can be obtained when the K\L{}
exponent is known. Specifically, under the one-sided K\L{} condition with
exponent $\theta\in(0,1)$, DS-GDA converges with an iteration complexity of
$\mathcal{O}(\epsilon^{-2\max\{2\theta,1\}})$. They all match the corresponding
best results in the literature. Moreover, we show that DS-GDA is practically
applicable to general nonconvex-nonconcave problems even without any regularity
conditions, such as the P\L{} condition, K\L{} condition, or weak Minty
variational inequalities condition. For various challenging
nonconvex-nonconcave examples in the literature, including ``Forsaken'',
``Bilinearly-coupled minimax'', ``Sixth-order polynomial'', and ``PolarGame'',
the proposed DS-GDA can all get rid of limit cycles. To the best of our
knowledge, this is the first first-order algorithm to achieve convergence on
all of these formidable problems.
