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Concentration of First Hitting Times Under Additive Drift

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Abstract

Recent advances in drift analysis have given us better and better tools for understanding random processes, including the run time of randomized search heuristics. In the setting of multiplicative drift we do not only have excellent bounds on the expected run time, but also more general results showing the strong concentration of the run time. In this paper we investigate the setting of additive drift under the assumption of strong concentration of the “step size” of the process. Under sufficiently strong drift towards the goal we show a strong concentration of the hitting time. In contrast to this, we show that in the presence of small drift a Gambler’s-Ruin-like behavior of the process overrides the influence of the drift, leading to a maximal movement of about \(\sqrt{t}\) steps within t iterations. Finally, in the presence of sufficiently strong negative drift the hitting time is superpolynomial with high probability; this corresponds to the well-known Negative Drift Theorem.

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Notes

  1. Note that iterating this idea leads to an example where, under arbitrary additive drift, the expected number of iterations until n is reached is 2, seemingly contradicting the Additive Drift Theorem; however, this iterated example requires an unbounded search space, which is ruled out by the requirements of the Additive Drift Theorem.

  2. In fact, the term sub-Gaussian originally entailed \(\delta = \infty \), while the version with finite \(\delta \) is called locally sub-Gaussian [3]. Furthermore, this terminology is usually applied to single random variables, not to martingales.

  3. Note that Eq. (2) is typically required to also hold for negative \(z \ge -\delta \), in which case even stronger statements can be made. However, we want to keep the scope of these definitions wide.

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Acknowledgments

I would like to thank Tobias Friedrich, Benjamin Doerr and Anton Krohmer for many useful discussions on the topic of this paper; all of them also provided valuable pointers to the literature. A further source of pointers to the literature as well as many helpful comments came from the reviewers of the conference version this paper, who have proved to be very knowledgeable in this subject. Most importantly, Carsten Witt pointed me to possible extensions of the Azuma–Hoeffding Inequality, as well as to some computations in his own publications, which helped to extend this paper to cases of non-bounded variables, making the contribution significantly more valuable; this paper would have a much more restricted scope without his advice. Finally, I would like to thank the reviewers of the journal version of this paper; they pointed out places for correction and improvement, as well as more useful literature.

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  1. Fiedrich-Schiller-Universität Jena, 07743, Jena, Germany

    Timo Kötzing

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  1. Timo Kötzing
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Kötzing, T. Concentration of First Hitting Times Under Additive Drift. Algorithmica 75, 490–506 (2016). https://doi.org/10.1007/s00453-015-0048-0

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