Silver Rate Is (Almost) Optimal for Gradient Descent

We study how far gradient descent (GD) can be accelerated by predetermined stepsizes in smooth convex optimization. Writing $p_{\mathrm{s}\mathrm{i}\mathrm{l}}={\log }_2(1+\sqrt{2})$, we prove an $\mathrm{\Omega }\left(n^{-p_{\mathrm{s}\mathrm{i}\mathrm{l}}-O(\sqrt{\log \log n\mathrm{/}\log n})}\right)$ non-anytime lower bound. In the anytime setting, every infinite schedule has infinitely many horizons with error $\mathrm{\Omega }\left(n^{-\frac{2p_{\mathrm{s}\mathrm{i}\mathrm{l}}}{1+p_{\mathrm{s}\mathrm{i}\mathrm{l}}}-O(\sqrt{\log \log n\mathrm{/}\log n})}\right)$. 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.