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

## TL;DR

This paper provides tight upper bounds on the expected runtime of the Univariate Marginal Distribution Algorithm (UMDA) on the OneMax problem for different parameter regimes, improving upon previous bounds and complementing existing lower bounds.

## Contribution

It derives new upper bounds on UMDA's runtime for specific parameter ranges, showing tightness and extending understanding of its efficiency on OneMax.

## Key findings

- Expected runtime is O(μn) for μ ≥ c log n with confined marginals.
- Expected runtime is O(μ√n) for μ ≥ c' √n log n, often without borders.
- Results complement lower bounds and improve previous upper bounds.

## Abstract

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 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 $\mu\ge c' \sqrt{n}\log n$ for a constant $c'>0$ and $\lambda=(1+\Theta(1))\mu$, the behavior 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 marginal probabilities 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 as tight for the two very different values $\mu=c\log n$ and $\mu=c'\sqrt{n}\log n$. They also improve the previously best known upper bound $O(n\log n\log\log n)$ by Dang and Lehre (GECCO 2015).