Silver Rate Is (Almost) Optimal for Gradient Descent: The Strongly Convex Case

2026-09-22

AI Summary

研究证明了在光滑强凸函数上，使用预定的非负步长进行梯度下降时，Silver步长方案几乎是最优的。

Abstract

We study gradient descent with predetermined nonnegative stepsizes on smooth strongly convex functions. Let $p_{\mathrm{sil}}=\log_2(1+\sqrt2)$ and $κ$ be the condition number. We prove the iteration lower bound $Ω\left(κ^{\frac{1}{p_{\mathrm{sil}}}-o(1)}\log\frac1δ\right)$ for both relative squared distance and relative function error, uniformly over $0<δ<1$ and sufficiently large $κ$. This matches the polynomial exponent of $κ$ for the Silver stepsize schedule established in [Altschuler and Parrilo, 2025].

Problem

• Gradient Descent
• Strongly Convex Functions
• Iteration Lower Bound
• Condition Number

Innovation

• gradient descent
• strongly convex functions
• Silver stepsize
• iteration lower bound
• condition number