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

Abstract: 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 $Ω(κLΔσ<sup>2ε<sup>{-4})$ under the bounded-variance assumption and $Ω(κ^{3/2}\bar LΔσε<sup>{-3})$ under the additional assumption of averaged smoothness. Here, $L$ and $\bar L$ denote the smoothness and averaged-smoothness constants, respectively, $Δ$ is the initial primal gap, $σ<sup>2$ bounds the oracle variance, and $κ=L/μ$ or $\bar L/μ$ in the respective settings, where $μ$ is the strong-concavity parameter. Our bounded-variance lower bound improves the dependence on the condition number from $κ<sup>{1/3}$ in previous lower bounds to $κ$, 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 $κ$ 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.