Complexity Lower Bounds for Nonconvex-Strongly-Concave Min-Max Optimization

04/18/2021 ∙ by Haochuan Li, et al. ∙ 10

We provide a first-order oracle complexity lower bound for finding stationary points of min-max optimization problems where the objective function is smooth, nonconvex in the minimization variable, and strongly concave in the maximization variable. We establish a lower bound of Ω(√(κ)ϵ^-2) for deterministic oracles, where ϵ defines the level of approximate stationarity and κ is the condition number. Our analysis shows that the upper bound achieved in (Lin et al., 2020b) is optimal in the ϵ and κ dependence up to logarithmic factors. For stochastic oracles, we provide a lower bound of Ω(√(κ)ϵ^-2 + κ^1/3ϵ^-4). It suggests that there is a significant gap between the upper bound 𝒪(κ^3 ϵ^-4) in (Lin et al., 2020a) and our lower bound in the condition number dependence.

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