Asymptotic Lower Bounds for Continuous Optimization

Abstract: Nonasymptotic convergence rates for optimization problems have been extensively studied across a wide range of settings, with carefully constructed worst-case instances establishing matching lower bounds for many algorithm classes. Recent work has shown, however, that these rates can often be improved in the asymptotic regime, while the corresponding asymptotic lower bounds remain largely unknown. We propose a general technique for converting existing nonasymptotic lower bound constructions into asymptotic lower bounds on Hilbert spaces. This allows us to show that several known asymptotic convergence upper bounds in Hilbert spaces are tight. In particular, we recover tightness of the $o(n<sup>{-2})$ suboptimality guarantee for accelerated gradient descent on smooth convex functions \cite{attouch2016rate}, as well as the $o(n<sup>{-1/2})$ suboptimality guarantee for gradient descent on smooth nonconvex functions \cite{gratton2025refining}. For stochastic convex optimization, we also develop a method that achieves an $o(n<sup>{-1/2})$ suboptimality guarantee in finite dimensions, and prove a matching one-dimensional lower bound.