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

Abstract

We study gradient descent with predetermined nonnegative stepsizes on smooth strongly convex functions. Let $p_{\mathrm{sil}}=\log_2(1+\sqrt2)$ and $\kappa$ be the condition number. We prove the iteration lower bound $\Omega\left(\kappa^{\frac{1}{p_{\mathrm{sil}}}-o(1)}\log\frac1\delta\right)$ for both relative squared distance and relative function error, uniformly over $0<\delta<1$ and sufficiently large $\kappa$. This matches the polynomial exponent of $\kappa$ for the Silver stepsize schedule established in [Altschuler and Parrilo, 2025].