Determine asymptotic lower bounds for improved asymptotic convergence rates

Determine the asymptotic lower bounds corresponding to convergence rates for continuous optimization methods that improve upon their nonasymptotic rates in the asymptotic regime.

Background

The paper studies the gap between nonasymptotic convergence guarantees and sharper asymptotic rates in continuous optimization. Although asymptotic upper bounds have recently improved in several settings, matching lower bounds are not generally established. The paper develops a construction that resolves this issue for several deterministic and stochastic optimization classes, including accelerated gradient descent on smooth convex functions, gradient descent on smooth nonconvex functions, and stochastic convex optimization; the broader problem remains unresolved beyond these settings.