Mean-performance of Sharp Restart II: Inequality Roadmap

TL;DR

This paper introduces an inequality-based framework to analyze how deterministic restart strategies affect the mean completion time of stochastic processes, revealing that restart's impact depends on the process's statistical heterogeneity.

Contribution

The paper develops a novel inequality indices approach to characterize when sharp restart accelerates or delays process completion, providing universal criteria based on heterogeneity measures.

Findings

Restart impedes mean completion when heterogeneity is low.

Restart expedites mean completion when heterogeneity is high.

Universal inequality criteria determine restart effects across processes.

Abstract

Restarting a deterministic process always impedes its completion. However, it is known that restarting a random process can also lead to an opposite outcome -- expediting completion. Hence, the effect of restart is contingent on the underlying statistical heterogeneity of the process' completion times. To quantify this heterogeneity we bring a novel approach to restart: the methodology of inequality indices, which is widely applied in economics and in the social sciences to measure income and wealth disparity. Using this approach we establish an `inequality roadmap' for the mean-performance of sharp restart: a whole new set of universal inequality criteria that determine when restart with sharp timers (i.e. with fixed deterministic timers) decreases/increases mean completion. The criteria are based on a host of inequality indices including Bonferroni, Gini, Pietra, and other Lorenz-curve indices; each index captures a different angle of the restart-inequality interplay. Utilizing the fact that sharp restart can match the mean-performance of any general restart protocol, we prove -- with unprecedented precision and resolution -- the validity of the following statement: restart impedes/expedites mean completion when the underlying statistical heterogeneity is low/high.