Optimal Recursive Composition and Dyadic Phase Laws for Gradient Descent with Predetermined Stepsizes

RESUMO

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 non-constant log-periodic modulation across prescribed horizons. Specifically, for symmetric recursive frameworks (primitive and OBS-S constructions), we prove that for every $N \geq 1$, the corresponding optimized schedules satisfy $f(x_{N-1})-f^\ast \le \frac{1}{2N^p \Phi(\log_2N)-1} \frac{L}{2}\|x_0-x^\ast\|^2$, $ p=\log_2(1+\sqrt2)$, where $\Phi$ 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 the asymmetric framework (the OBS-F construction), we show that although optimal splits are not necessarily balanced, the same Silver exponent asymptotically persists alongside a distinct log-periodic modulation.