Upper Bounds on the Running Time of the Univariate Marginal Distribution Algorithm on OneMax

The Univariate Marginal Distribution Algorithm (UMDA) is a randomized search heuristic that builds a stochastic model of the underlying optimization problem by repeatedly sampling $$\lambda $$λ solutions and adjusting the model according to the best $$\mu $$μ samples. We present a running time analysis of the UMDA on the classical OneMax benchmark function for wide ranges of the parameters $$\mu $$μ and $$\lambda $$λ. If $$\mu \ge c\log n$$μ≥clogn for some constant $$c>0$$c>0 and $$\lambda =(1+\varTheta (1))\mu $$λ=(1+Θ(1))μ, we obtain a general bound $$O(\mu n)$$O(μn) on the expected running time. This bound crucially assumes that all marginal probabilities of the algorithm are confined to the interval $$[1/n,1-1/n]$$[1/n,1-1/n]. If $$\mu \ge c' \sqrt{n}\log n$$μ≥c′nlogn for a constant $$c'>0$$c′>0 and $$\lambda =(1+\varTheta (1))\mu $$λ=(1+Θ(1))μ, the behavior of the algorithm changes and the bound on the expected running time becomes $$O(\mu \sqrt{n})$$O(μn), which typically holds even if the borders on the marginal probabilities are omitted. The results supplement the recently derived lower bound $$\varOmega (\mu \sqrt{n}+n\log n)$$Ω(μn+nlogn) by Krejca and Witt (Proceedings of FOGA 2017, ACM Press, New York, pp 65–79, 2017) and turn out to be tight for the two very different choices $$\mu =c\log n$$μ=clogn and $$\mu =c'\sqrt{n}\log n$$μ=c′nlogn. They also improve the previously best known upper bound $$O(n\log n\log \log n)$$O(nlognloglogn) by Dang and Lehre (Proceedings of GECCO ’15, ACM Press, New York, pp 513–518, 2015) that was established for $$\mu =c\log n$$μ=clogn and $$\lambda =(1+\varTheta (1))\mu $$λ=(1+Θ(1))μ.