Diversity of Sharp Restart

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 restartcan 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.