aiqbench analysis
This paper establishes the theoretical limits of gradient descent acceleration using fixed step sizes, proving that the "silver rate" is essentially optimal for smooth convex optimization problems. By deriving tight non-anytime lower bounds and matching them with existing upper bounds, the study clarifies the fundamental convergence exponents achievable without adaptive scheduling mechanisms. These findings are critical for understanding the intrinsic efficiency boundaries of standard optimization algorithms used in large-scale machine learning training.
Silver Rate Is (Almost) Optimal for Gradient Descent Acceleration
arXiv:2609.09152v1 Announce Type: cross Abstract: We study how far gradient descent (GD) can be accelerated by predetermined nonnegative stepsizes in smooth convex optimization. Writing $p_{\mathrm{sil}}=\log_2(1+\sqrt{2})$, we prove an $\Omega\left(n^{-p_{\mathrm{sil}}-O(\sqrt{\log\log n/\log n})}\right)$ non-anytime lower bound. In the anytime setting, every infinite nonnegative schedule has infinitely many horizons with error $\Omega\left(n^{-\frac{2p_{\mathrm{sil}}}{1+p_{\mathrm{sil}}}-O(\sqrt{\log\log n/\log n})}\right)$. Together with the silver-schedule upper bound [Altschuler and Parrilo, 2025] and the anytime upper bound [Zhang et al., 2025], our results determine the optimal polynomial convergence exponents in both settings.
Abstract:We study how far gradient descent (GD) can be accelerated by predetermined nonnegative stepsizes in smooth convex optimization. Writing $p_{\mathrm{sil}}=\log_2(1+\sqrt{2})$, we prove an $\Omega\left(n^{-p_{\mathrm{sil}}-O(\sqrt{\log\log n/\log n})}\right)$ non-anytime lower bound. In the anytime setting, every infinite nonnegative schedule has infinitely many horizons with error $\Omega\left(n^{-\frac{2p_{\mathrm{sil}}}{1+p_{\mathrm{sil}}}-O(\sqrt{\log\log n/\log n})}\right)$. Together with the silver-schedule upper bound [Altschuler and Parrilo, 2025] and the anytime upper bound [Zhang et al., 2025], our results determine the optimal polynomial convergence exponents in both settings.
Optimization and Control (math.OC); Machine Learning (cs.LG)
arXiv:2609.09152 [math.OC]
(or arXiv:2609.09152v1 [math.OC] for this version)
https://doi.org/10.48550/arXiv.2609.09152 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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Full text under the publisher's open licence · 174 words
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