AI 中文总结
本文研究预定步长梯度下降的递归组合调度，证明对称框架中平衡分割最优并刻画对数周期收敛率，解决Zhang-Jiang猜想，并推广至非对称框架。
AI 中文摘要
最近研究表明，包含精心选择的长步长的预定步长调度可以加速光滑凸函数上的梯度下降（GD）。这类调度的一个突出类别是通过递归组合构建的。本文刻画了这些优化递归调度的收敛性，揭示了在指定视界上非恒定的对数周期调制。具体而言，对于对称递归框架（原始和OBS-S构造），我们证明对于每个$N \geq 1$，相应的优化调度满足$f(x_{N-1})-f^\ast \le \frac{1}{2N^p \Phi(\log_2N)-1} \frac{L}{2}\\|x_0-x^\ast\\|^2$，其中$p=\log_2(1+\sqrt2)$，$\Phi$是一个正的、Lipschitz连续的、非恒定的$1$-周期函数。我们通过证明在这些构造中平衡分割在每个视界上都是最优的来推导此结果，从而解决了Zhang和Jiang的一个猜想。此外，对于非对称框架（OBS-F构造），我们表明尽管最优分割不一定平衡，但相同的Silver指数渐近地持续存在，并伴随不同的对数周期调制。
英文摘要
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 $f(x_{N-1})-f^\ast \le \frac{1}{2N^p Φ(\log_2N)-1} \frac{L}{2}\|x_0-x^\ast\|^2$, $ 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.
