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 sil= 2(1+ sqrt 2), we prove an Ω (n-p sil-O(sqrt n/ n)) non-anytime lower bound. In the anytime setting, every infinite schedule has infinitely many horizons with error Ω (n-2p sil over 1+p sil-O(sqrt n/ n)). 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.