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

We study gradient descent with predetermined nonnegative stepsizes on smooth strongly convex functions. Let $p_{\mathrm{s}\mathrm{i}\mathrm{l}}={\log }_2(1+\sqrt{2})$ and $\kappa$ be the condition number. We prove the iteration lower bound $\mathrm{\Omega }({\kappa }^{\frac{1}{p_{\mathrm{s}\mathrm{i}\mathrm{l}}}-o(1)}\log \frac{1}{\delta })$ 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].