General Drift Analysis with Tail Bounds

Drift analysis is one of the state-of-the-art techniques for the runtime analysis of randomized search heuristics. In recent years, many different drift theorems, including additive, multiplicative and variable drift, have been developed, applied and partly generalized or adapted to particular processes. A comprehensive overview article was missing. We provide not only such an overview but also present a universal drift theorem that generalizes virtually all existing drift theorems found in the literature. On the one hand, the new theorem bounds the expected first hitting time of optimal states in the underlying stochastic process. On the other hand, it also allows for general upper and lower tail bounds on the hitting time, which were not known before except for the special case of upper bounds in multiplicative drift scenarios. As a proof of concept, the new tail bounds are applied to prove very precise sharp-concentration results on the running time of the (1+1) EA on OneMax, general linear functions and LeadingOnes. Moreover, user-friendly specializations of the general drift theorem are given.