Diversity of Sharp Restart

TL;DR

This paper investigates how sharp restart protocols influence the diversity of completion times in stochastic processes, providing universal criteria for when such protocols increase or decrease diversity, and analyzing their joint effects on mean and randomness.

Contribution

It introduces a diversity-based analysis of sharp restart effects, establishing universal criteria and practical conditions for controlling the diversity of completion times.

Findings

Sharp restart can both increase and decrease diversity of completion times.

Universal criteria determine when diversity is affected by sharp restart.

Joint effects on mean and diversity can be aligned or antithetical.

Abstract

When applied to a stochastic process of interest, a restart protocol alters the overall statistical distribution of the process' completion time; thus, the completion-time's mean and randomness change. The explicit effect of restart on the mean is well understood, and it is known that: from a mean perspective, deterministic restart protocols -- termed sharp restart -- can out-perform any other restart protocol. However, little is known on the explicit effect of restart on randomness. This paper is the second in a duo exploring the effect of sharp restart on randomness: via a Boltzmann-Gibbs-Shannon entropy analysis in the first part, and via a diversity analysis in this part. Specifically, gauging randomness via diversity -- a measure that is intimately related to the Renyi entropy -- this paper establishes a set of universal criteria that determine: A) precisely when a sharp-restart protocol decreases/increases the diversity of completion times; B) the very existence of sharp-restart protocols that decrease/increase the diversity of completion times. Moreover, addressing jointly mean-behavior and randomness, this paper asserts and demonstrates when sharp restart has an aligned effect on the two (decreasing/increasing both), and when the effect is antithetical (decreasing one while increasing the other). The joint mean-diversity results require remarkably little information regarding the (original) statistical distributions of completion times, and are remarkably practical and easy to implement.