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Tail bounds on hitting times of randomized search heuristics using variable drift analysis

Published online by Cambridge University Press:  05 November 2020

P. K. Lehre
Affiliation:
School of Computer Science, University of Birmingham, Birmingham B15 2TT, UK
C. Witt*
Affiliation:
DTU Compute, Technical University of Denmark, Kongens Lyngby, Denmark
*
*Corresponding author. Email: cawi@dtu.dk

Abstract

Drift analysis is one of the state-of-the-art techniques for the runtime analysis of randomized search heuristics (RSHs) such as evolutionary algorithms (EAs), simulated annealing, etc. The vast majority of existing drift theorems yield bounds on the expected value of the hitting time for a target state, for example the set of optimal solutions, without making additional statements on the distribution of this time. We address this lack by providing a general drift theorem that includes bounds on the upper and lower tail of the hitting time distribution. The new tail bounds are applied to prove very precise sharp-concentration results on the running time of a simple EA on standard benchmark problems, including the class of general linear functions. On all these problems, the probability of deviating by an r-factor in lower-order terms of the expected time decreases exponentially with r. The usefulness of the theorem outside the theory of RSHs is demonstrated by deriving tail bounds on the number of cycles in random permutations. All these results handle a position-dependent (variable) drift that was not covered by previous drift theorems with tail bounds. Finally, user-friendly specializations of the general drift theorem are given.

Type
Paper
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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Footnotes

A preliminary version of this paper appeared in the proceedings of ISAAC 2014 [28].

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