Improved Gradient Descent Lower Bounds Beyond Nesterov

Abstract:We study how far gradient descent (GD) can be accelerated by predetermined stepsizes in smooth convex optimization. Going beyond the classical $\Omega(n^{-2})$ first-order oracle lower bound of Nemirovsky and Yudin, we prove an $\Omega(n^{-1.6342})$ non-anytime lower bound and an $\Omega(n^{-1.2408})$ anytime lower bound. These improve the recent $\Omega(n^{-1.932})$ non-anytime lower bound of Ma and Chen and the $\Omega(n^{-4/3})$ 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.

Optimization and Control (math.OC); Machine Learning (cs.LG); Machine Learning (stat.ML)

arXiv:2609.02855 [math.OC]

(or arXiv:2609.02855v1 [math.OC] for this version)

https://doi.org/10.48550/arXiv.2609.02855 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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