Upper Bounds on the Runtime of the Univariate Marginal Distribution Algorithm on OneMax

A runtime analysis of the Univariate Marginal Distribution Algorithm (UMDA) is presented on the OneMax function for wide ranges of its parameters $μ$ and $λ$. If $μ\ge c\log n$ for some constant $c>0$ and $λ=(1+Θ(1))μ$, a general bound $O(μn)$ on the expected runtime is obtained. This bound crucially assumes that all marginal probabilities of the algorithm are confined to the interval $[1/n,1-1/n]$. If $μ\ge c' \sqrt{n}\log n$ for a constant $c'>0$ and $λ=(1+Θ(1))μ$, the behavior of the algorithm changes and the bound on the expected runtime becomes $O(μ\sqrt{n})$, which typically even holds if the borders on the marginal probabilities are omitted. The results supplement the recently derived lower bound $Ω(μ\sqrt{n}+n\log n)$ by Krejca and Witt (FOGA 2017) and turn out as tight for the two very different values $μ=c\log n$ and $μ=c'\sqrt{n}\log n$. They also improve the previously best known upper bound $O(n\log n\log\log n)$ by Dang and Lehre (GECCO 2015).