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

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 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 $$f(x_{N-1})-f^\ast \le \frac{1}{2N^p Φ(\log_2N)-1} \frac{L}{2}|x_0-x<sup>\ast|<sup>2$$, $p=\log_2(1+\sqrt2)$, 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 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.