A lower bound for stepsize-based acceleration of gradient descent

• URL: http://arxiv.org/abs/2608.10418v1
• Date: Tue, 11 Aug 2026 03:06:59 GMT
• Title: A lower bound for stepsize-based acceleration of gradient descent
• Abstract summary: We present a new lower bound of $(T-1.9319)$ for the last-iterate convergence rate of gradient descent with predetermined nonnegative stepsize schedules.<n>This result provides rigorous evidence that stepsize schedules alone cannot accelerate plain GD to the optimal $O(T-2)$ convergence rate.
• Score: 18.733838876040537
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Recent work has shown that, for smooth convex optimization, plain gradient descent can be accelerated from its textbook convergence rate of $O(T^{-1})$ (where $T$ denotes the number of iterations) to $O\big(T^{-\log_2(1+\sqrt{2})}\big)$ using carefully designed stepsize schedules alone, without resorting to momentum or other algorithmic modifications. Despite this progress, however, little was known about lower bounds for such methods beyond the classical $Ω(T^{-2})$ benchmark for general first-order methods. In this work, we present a new lower bound of $Ω(T^{-1.9319})$ for the last-iterate convergence rate of gradient descent with predetermined nonnegative stepsize schedules. This result provides rigorous evidence that stepsize schedules alone cannot accelerate plain GD to the optimal $O(T^{-2})$ convergence rate. The proof was developed by GPT-5.6 Sol Pro under the authors' guidance.

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