Stochastic Gradient Descent Ascent is Suboptimal for Nonconvex-PL Min-Max Games

arXiv:2610.07814v1 Announce Type: cross Abstract: How far can stochastic gradient descent ascent (SGDA) go by tuning its timescale ratio and step sizes in nonconvex min-max games? We answer this question for nonconvex-PL (NC-PL) games by establishing the first tight complexity of two-timescale SGDA with a fixed timescale ratio and non-increasing step sizes. For $\ell$-smooth games with an inner $\mu$-PL inequality, we prove a complexity lower bound $\Omega(\kappa^2\ell\varepsilon^{-2}+\kappa^4\ell\sigma^2\varepsilon^{-4})$, where $\kappa=\ell/\mu$ is the condition number, $\sigma^2$ is the gradient variance, and $\varepsilon$ measures the outer gradient norm. This matches existing SGDA upper bounds and establishes a complexity separation from Smoothed-AGDA (Yang et al., 22'). In addition, we show that SGDA can fail to find a stationary point when its timescale ratio is as small as $o(\kappa^2)$. Our negative results highlight the fundamental limitation of SGDA in NC-PL games, and justify the development of alternative methods.