Entropy of Sharp Restart

TL;DR

This paper investigates how sharp restart protocols influence the entropy of completion times in stochastic processes, providing closed-form formulas and conditions for entropy increase or decrease, thus illuminating the interplay between restart strategies and randomness.

Contribution

It introduces the first comprehensive analysis of how sharp restart affects the Boltzmann-Gibbs-Shannon entropy of completion times, including closed-form results and geometric conditions.

Findings

Closed-form formulas for entropy under sharp restart with general timers.

Identification of conditions where restart increases or decreases entropy.

Comparison of hazard rates to a flat exponential benchmark.

Abstract

Restart has the potential of expediting or impeding the completion times of general random processes. Consequently, the issue of mean-performance takes center stage: quantifying how the application of restart on a process of interest impacts its completion-time's mean. Going beyond the mean, little is known on how restart affects stochasticity measures of the completion time. This paper is the first in a duo of studies that address this knowledge gap via: a comprehensive analysis that quantifies how sharp restart -- a keystone restart protocol -- impacts the completion-time's Boltzmann-Gibbs-Shannon entropy. The analysis establishes closed-form results for sharp restart with general timers, with fast timers (high-frequency resetting), and with slow timers (low-frequency resetting). These results share a common structure: comparing the completion-time's hazard rate to a flat benchmark -- the constant hazard rate of an exponential distribution whose entropy is equal to the completion-time's entropy. In addition, using an information-geometric approach based on Kullback-Leibler distances, the analysis establishes results that determine the very existence of timers with which the application of sharp restart decreases or increases the completion-time's entropy. Our work sheds first light on the intricate interplay between restart and randomness -- as gauged by the Boltzmann-Gibbs-Shannon entropy.