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

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