General Drift Analysis with Tail Bounds

TL;DR

This paper introduces a comprehensive drift theorem that provides bounds on the distribution tails of hitting times, enabling precise runtime concentration results for randomized algorithms and extending to various applications beyond optimization.

Contribution

It presents a general drift theorem with tail bounds applicable to position-dependent drifts, unifying and extending existing drift theorems with new concentration results.

Findings

Exponential decay of deviation probability in running time for simple EAs.

Sharp concentration results for linear functions in benchmark problems.

Tail bounds for the number of cycles in random permutations.

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, e.g., 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. Surprisingly, the probability of deviating by an $r$-factor in lower order terms of the expected time decreases exponentially with $r$ on all these problems. 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. Moreover, our theorem can be specialized into virtually all existing drift theorems with drift towards the target from the literature. Finally, user-friendly specializations of the general drift theorem are given.