TY - JOUR

T1 - Diversity of sharp restart

AU - Eliazar, Iddo

AU - Reuveni, Shlomi

N1 - Publisher Copyright:
© 2023 The Author(s). Published by IOP Publishing Ltd.

PY - 2023/1/13

Y1 - 2023/1/13

N2 - 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 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.

AB - 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 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.

KW - diversity

KW - low-frequency and high-frequency resetting

KW - Renyi entropy

KW - restart protocols

KW - sharp restart

UR - http://www.scopus.com/inward/record.url?scp=85146914343&partnerID=8YFLogxK

U2 - 10.1088/1751-8121/acb184

DO - 10.1088/1751-8121/acb184

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AN - SCOPUS:85146914343

SN - 1751-8113

VL - 56

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

IS - 2

M1 - 024003

ER -