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

Abstract

We study the stochastic first-order oracle complexity of finding $\epsilon$-stationary points of the primal function in smooth nonconvex-strongly-concave minimax optimization. For sufficiently small $\epsilon$, we establish lower bounds of $\Omega(\kappa L\Delta\sigma^2\epsilon^{-4})$ under the bounded-variance assumption and $\Omega(\kappa^{3/2}\bar L\Delta\sigma\epsilon^{-3})$ 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 $\epsilon$. 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.