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

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 −2 ) for deterministic oracles, where defines the level of approximate stationarity and κ is the condition number. Our lower bound matches the best existing upper bound in the and κ dependence up to logarithmic factors. For stochastic oracles, we provide a lower bound of Ω (√ κ −2 + κ −4 ) . It suggests that there is a gap between the best existing upper boundO(κ −4) and our lower bound in the condition number dependence.

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