连续优化的渐近下界
Asymptotic Lower Bounds for Continuous Optimization
- University of Pittsburgh（匹兹堡大学）
- Shanghai University of Finance and Economics（上海财经大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出将非渐近下界转化为希尔伯特空间渐近下界的通用技术，证明加速梯度下降和梯度下降的已知渐近上界是紧的，并为随机凸优化建立了匹配下界。
AI中文摘要：
优化问题的非渐近收敛速率已在广泛场景中被深入研究，通过精心构造的最坏情况实例，为许多算法类别建立了匹配的下界。然而，近期研究表明，这些速率在渐近情形下通常可以得到改进，而相应的渐近下界在很大程度上仍然未知。我们提出了一种通用技术，将现有的非渐近下界构造转化为希尔伯特空间上的渐近下界。这使得我们能够证明希尔伯特空间中若干已知的渐近收敛上界是紧的。特别地，我们恢复了光滑凸函数上加速梯度下降的$o(n^{-2})$次优性保证的紧性\cite{attouch2016rate}，以及光滑非凸函数上梯度下降的$o(n^{-1/2})$次优性保证的紧性\cite{gratton2025refining}。对于随机凸优化，我们还开发了一种在有限维中实现$o(n^{-1/2})$次优性保证的方法，并证明了一个匹配的一维下界。
英文摘要：
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 $o(n^{-2})$ suboptimality guarantee for accelerated gradient descent on smooth convex functions \cite{attouch2016rate}, as well as the $o(n^{-1/2})$ suboptimality guarantee for gradient descent on smooth nonconvex functions \cite{gratton2025refining}. For stochastic convex optimization, we also develop a method that achieves an $o(n^{-1/2})$ suboptimality guarantee in finite dimensions, and prove a matching one-dimensional lower bound.
