急剧重启的平均性能I：统计路线图
Mean-performance of sharp restart I: Statistical roadmap
摘要：
重新启动是一个具有重要意义和广泛适用性的加速一般随机过程的首次通过时间和完成时间的一般框架。重新启动协议可以使用确定性或随机计时器。具有确定性定时器的重启协议--“锐重启”--承担主要角色：如果存在提高平均性能的重启协议，则存在性能良好或更好的锐重启协议。本文，一对二人组的第一个，提出了一个全面的平均性能分析急剧重启。使用统计方法，该分析建立了确定急剧重启何时改善或降低平均性能的通用标准，即，减少或增加平均首次通过/完成时间。这些标准类似于最近发现的最广泛应用的重启协议-“指数重启”-使用指数分布的计时器。然而，虽然指数重启标准仅涵盖慢定时器的情况，但此处建立的急剧重启标准进一步涵盖快速、关键和一般定时器的情况;此外，后一标准解决了急剧重启改善或降低平均性能的定时器的存在。使用慢计时器的标准，我们发现了一个一般的情况下，其中：急剧重启提高平均性能，而指数重启的平均性能。文中给出的新结果的有效性通过例子和结果在正则扩散过程中的应用得到了证明。
英文摘要：
Restart is a general framework, of prime importance and wide applicability, for expediting first-passage times and completion times of general stochastic processes. Restart protocols can use either deterministic or stochastic timers. Restart protocols with deterministic timers -- "sharp restart" -- assume a principal role: if there exists a restart protocol that improves mean-performance, then there exists a sharp-restart protocol that performs as good or better. This paper, the first of a duo, presents a comprehensive mean-performance analysis of sharp restart. Using statistical methods, the analysis establishes universal criteria that determine when sharp restart improves or worsens mean-performance, i.e., decreases or increases mean first-passage/completion times. These criteria are akin to those recently discovered for the most widely applied restart protocols -- "exponential restart" -- which use exponentially-distributed timers. However, while the exponential-restart criteria cover only the case of slow timers, the sharp-restart criteria established here further cover the cases of fast, critical, and general timers; moreover, the latter criteria address the very existence of timers with which sharp restart improves or worsens mean-performance. Using the slow-timers criteria, we discover a general scenario for which: sharp restart improves mean-performance, whereas exponential restart worsens mean-performance. The potency of the novel results presented here is demonstrated by examples, and by the results' application to canonical diffusion processes.
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