Optimal Recursive Composition and Dyadic Phase Laws for Gradient Descent with Predetermined Stepsizes
Year: 2026
Type: preprint
Abstract: Predetermined stepsize schedules featuring carefully chosen long steps have recently been shown to accelerate gradient descent (GD) on smooth convex functions. A prominent class of such schedules is built through recursive composition. In this paper, we characterize the convergence of these optimized recursive schedules, revealing a nonconstant log-periodic modulation across prescribed horizons. Specifically, we prove that for every $N \geq 1$, the optimized ConPP/OBS-S schedules satisfy $$f(x_{N-1})-f_\ast \le \frac{1}{2N^pΦ(\log_2N)-1} \frac{L}{2}\|x_0-x_\ast\|^2, \qquad p=\log_2(1+\sqrt2), $$ where $Φ$ is a positive, Lipschitz, nonconstant $1$-periodic function. We derive this by proving that balanced splitting is optimal at every horizon for these constructions, resolving a conjecture of Zhang and Jiang. Furthermore, for their asymmetric counterparts (ConPD/OBS-F schedules), we show that although optimal splits are not necessarily balanced, the same Silver exponent asymptotically persists alongside a distinct log-periodic modulation.
Source: arXiv (Cornell University)
Authors: Liu, Yu, Chen, Kang, Jiang, Rujun, Wang, Tianyu
Institution: Fudan University
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7212259628