For smooth functions, one natural idea is to use gradient descent ascent (GDA) [41], which applies gradient descent on the primal function and gradient ascent on the dual function. However, GDA is originally designed for the strongly-convex-strongly-concave problem, where either primal or dual players can dominate the game. When applying it to the nonconvex-concave case, a so-called two-timescale method can make it converge. In this scenario, the primal player is the dominant player in the game. We can regard this one-step GDA scheme as an inexact subgradient descent on the inner max function  $\max_{y\in \mathcal{Y}}f(x,y)$ , thus it is necessary for the dual update to be relatively faster than the primal update at each iteration. However, this two-timescale GDA yields a high iteration complexity of  $\mathcal{O}(\epsilon^{-6})$ . To achieve a lower iteration complexity, smoothed GDA (S-GDA) [62, 59] employs the Moreau-Yosida smoothing technique to stabilize the primal sequence. The resulting stabilized sequence can help S-GDA achieve a lower iteration complexity of  $\mathcal{O}(\epsilon^{-4})$ . Alternating Gradient Projection (AGP) [57] can also achieve this lower iteration complexity by adding regularizers to both primal and dual functions. It is worth noting that the convergence of all these algorithms is heavily contingent on the convexity/concavity of primal/dual functions, leading to asymmetric updates. In the convex-nonconcave scenario, the roles are reversed, and the dual player takes dominance in the game. To apply the aforementioned algorithms, their updates should be modified accordingly. Specifically, for SGDA and AGP, the smoothing and regularized sides also need to be changed to guarantee the optimal convergence rate. However, when dealing with nonconvex-nonconcave problems, no player inherently dominates the other and all existing first-order algorithms cannot be guaranteed to converge to stationary points (see Definition 1), and they can even suffer from limit cycles. That is, the generated trajectories of all these algorithms will converge to cycling orbits that do not contain any stationary point of  $f$ . Such spurious convergence phenomena arise from the inherent minimax structure of (P) and have no counterpart in pure minimization problems. Conceptually, nonconvex-nonconcave minimax optimization problems can be understood as a seesaw game, which means no player inherently dominates the other. More explicitly, the key difficulty lies in adjusting the primal and dual updates to achieve a good balance. Most existing works address this challenge by adding additional regularity conditions to ensure the automatic domination of one player over the other. Specifically, research works either impose the global Polyak-Lojasiewicz (PL) condition on the dual function  $f(x,\cdot)$  [59, 47, 22, 58] or assume the satisfaction of  $\alpha$ -dominance condition [29, 30]. Both approaches fall into this line of research, enabling the adoption of algorithms and analysis for nonconvex-concave minimax problems. Although the recent work [40] extends the PL condition to the one-sided Kurdyka-Lojasiewicz (KL) property, it is still hard to provide the explicit KL exponent, and prior knowledge regarding which side satisfies the KL condition is required to determine the appropriate side to employ extrapolation. If we choose the wrong side, it will result in a slow convergence or even divergence (see Figure 5). On another front, variational inequality (VI) provides a unified framework for the study of equilibrium/minimax problems [46, 36, 26, 60, 45]. However, VI-related conditions are usually hard to check in practice. Hence, the convergence of the existing algorithms all highly rely on prior knowledge of the primal/dual function, which makes it paramount to design a universal algorithm for convex-nonconcave, nonconvex-concave, and nonconvex-nonconcave minimax problems.