Improved Gradient Descent Lower Bounds Beyond Nesterov
We study how far gradient descent (GD) can be accelerated by predetermined stepsizes in smooth convex optimization. Going beyond the classical $\Omega(n^{-2})$ first-order oracle lower bound of Nemirovsky and Yudin (1983), we prove an $\Omega(n^{-1.6342})$ non-anytime lower bound and an $\Omega(n^{-1.2408})$ anytime lower bound. These improve the recent $\Omega(n^{-1.932})$ non-anytime lower bound of Ma and Chen (2026) and the $\Omega(n^{-4/3})$ anytime lower bound of Tsai et al. (2026), respectively. Both results continue to hold when the stepsizes may be negative. Our anytime lower bound also shows that the $O(n^{-\log_2(1+\sqrt{2})})$ rate of non-anytime silver schedules (Altschuler and Parrilo, 2025; Grimmer et al., 2025) is unattainable in the anytime setting. This establishes a strict separation between the two settings.