Improved Gradient Descent Lower Bounds Beyond Nesterov
Yuhan Ye, Kaizhao Liu
math.OC cs.LG stat.ML
Abstract
We study how far gradient descent (GD) can be accelerated by predetermined stepsizes in smooth convex optimization. Going beyond the classical $Ω(n^{-2})$ first-order oracle lower bound of Nemirovsky and Yudin, we prove an $Ω(n^{-1.6342})$ non-anytime lower bound and an $Ω(n^{-1.2408})$ anytime lower bound. These improve the recent $Ω(n^{-1.932})$ non-anytime lower bound of Ma and Chen and the $Ω(n^{-4/3})$ anytime lower bound of Tsai et al., respectively. Together with the non-anytime $O(n^{-\log_2(1+\sqrt{2})})$ rate achieved by silver schedules, our anytime lower bound establishes a strict separation between the achievable convergence exponents in the two settings.
Topics
Classified with taxonomy v2 on Thu, 3 Sept 2026.