关于重球动力学的两个答案
Two Answers on Heavy-Ball Dynamics
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中文总结 AI 辅助
本文回答了重球方法在光滑强凸函数上的两个问题：确定了Polyak调参的维度一致最坏情况指数收敛速率，并刻画了任意步长和动量下的相图，通过角度条件与Farey分数间隙完整描述收敛与周期轨道行为。
中文摘要 AI 辅助
我们回答了光滑强凸函数上重球方法的两个问题。第一个问题涉及Polyak的调参。其维度一致的 worst-case 指数收敛速率等于可插值混合旋转几何序列的最大增长因子，我们为每个条件数确定了该因子。最优混合构成四个代数族；该分类结合了精确多项式证书、有理参数化、临界三循环处五次数域中的精确解以及验证的区间延拓。两个频率就足够，维度五中的一个函数在每个时间范围达到该速率，上下界相差常数因子，且该方法在整个类上收敛当且仅当 $\kappa<9+4\sqrt5$，该阈值由Badithela和Seiler定位。第二个问题涉及任意步长和动量，特别是参数平面中计算出的Lyapunov区域和循环区域之间的区域。我们通过两个极端二次型的特征多项式之间的角度 $\Theta(\omega)$ 描述相图。当 $\Theta(\omega)>\omega/2$ 时方法收敛，当 $\Theta(2\pi j/k)\le\pi/k$ 时存在 $k$ 周期轨道，其余区域是由Farey分数组织的旋转数间隙。单个增长指数处的严格证书界定了速率，因此可以在阈值指数处证明收敛。在间隙的第一层，累积在共振端点的Farey分数扇形携带具有多个谐波的周期轨道，显式的扇形滞后不等式证明收敛，并且这两种行为都发生在之前未解决的区域中。
英文摘要
We answer two questions about the heavy-ball method on smooth strongly convex functions. The first concerns Polyak's tuning. Its dimension-uniform worst-case exponential rate equals the largest growth factor of an interpolable mixture of rotating geometric sequences, and we determine this factor for every condition number. The optimal mixtures form four algebraic families; the classification combines exact polynomial certificates, a rational parametrization, an exact solution in a quintic number field at the critical three-cycle, and verified interval continuation. Two frequencies suffice, one function in dimension five attains the rate at every horizon, the upper and lower bounds differ by constant factors, and the method converges on the whole class if and only if $κ<9+4\sqrt5$, the threshold located by Badithela and Seiler. The second question concerns arbitrary step sizes and momenta, in particular the regions of the parameter plane between the computed Lyapunov and cycle regions. We describe the phase diagram through the angle $Θ(ω)$ between the characteristic polynomials of the two extreme quadratics. The method converges when $Θ(ω)>ω/2$, has a $k$-periodic orbit when $Θ(2πj/k)\leπ/k$, and the remaining regions are rotation-number gaps organized by Farey fractions. A single strict certificate at one growth exponent bounds the rate, so convergence can be certified at the threshold exponent. In the first level of the gaps, fans of Farey fractions accumulating at a resonant endpoint carry periodic orbits with several harmonics, explicit fan-lag inequalities certify convergence, and both behaviors occur in the regions left open before.
发表机构
- Mohamed bin Zayed University of Artificial Intelligence（穆罕默德·本·扎耶德人工智能大学）
机构由 AI 辅助整理，请以论文原文为准。
