Tight Lower Bounds for Stochastic Nonconvex–Strongly-Concave Minimax Optimization

We study the stochastic first-order oracle complexity of finding $ϵ$-stationary points of the primal function in smooth nonconvex-strongly-concave minimax optimization. For sufficiently small $ϵ$, we establish lower bounds of $\Omega (\kappa L\Delta {\sigma }^2{ϵ}^{-4})$ under the bounded-variance assumption and $\Omega ({\kappa }^{3/2}\bar{L}\Delta \sigma {ϵ}^{-3})$ under the additional assumption of averaged smoothness. Here, $L$ and $\bar{L}$ denote the smoothness and averaged-smoothness constants, respectively, $\Delta$ is the initial primal gap, ${\sigma }^2$ bounds the oracle variance, and $\kappa =L/\mu$ or $\bar{L}/\mu$ in the respective settings, where $\mu$ is the strong-concavity parameter. Our bounded-variance lower bound improves the dependence on the condition number from ${\kappa }^{1/3}$ in previous lower bounds to $\kappa$, while our averaged-smoothness lower bound is the first of its kind. In both settings, the resulting lower bounds match existing upper bounds in their dependence on $\kappa$ and $ϵ$. Our proofs are based on a unified quadratic lifting construction that transfers a hardness instance for stochastic nonconvex minimization to unconstrained minimax optimization while preserving the required variance and smoothness properties.