基于无源性的一阶动量法分析
A Passivity-Based Analysis of First-Order Momentum-Based Methods
AI总结：
本文针对梯度具扇区界的函数，通过无源性分析推导一阶动量法的超参数条件，证明其迭代可收敛到全局极小值点。
AI中文摘要：
本文针对一类梯度具有下界0和上界L扇区界的函数，对一阶动量法开展离散时间无源性分析。通过循环变换证明，动量法可表示为与输出严格无源（OSP）系统负反馈连接的无源控制器。随后利用弱无源性定理推导超参数的显式条件，确保移位梯度渐近消失。在额外假设（存在唯一驻点且排除远离该点的任意小梯度）下，证明迭代序列收敛到全局极小值点。
英文摘要：
This paper presents a discrete-time passivity-based analysis of first-order momentum-based methods for a class of functions whose gradient has lower and upper sector bounds of $0$ and $L$, respectively. Through a loop transformation, it is shown that momentum-based methods can be represented as a passive controller in negative feedback with an output strictly passive (OSP) system. The weak passivity theorem is then used to derive explicit hyperparameter conditions under which the shifted gradient asymptotically vanishes. Under an additional assumption that requires the existence of a unique stationary point and excludes arbitrarily small gradients far from that point, convergence of the iterates to the global minimizer is established.
