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 $\mu$ 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).Comment: Version 4: added illustrations and experiments; improved presentation in Section 2.2; to appear in Algorithmica; the final publication is available at Springer via http://dx.doi.org/10.1007/s00453-018-0463-

Average time complexity of estimation of distribution algorithms

This paper presents a study based on the empirical results of the average first hitting time of Estimation of Distribution Algorithms. The algorithms are applied to one example of linear, pseudo-modular, and unimax functions. By means of this study, the paper also addresses recent issues in Estimation of Distribution Algorithms: (i) the relationship between the complexity of the probabilistic model used by the algorithm and its efficiency, and (ii) the matching between this model and the relationship among the variables of the objective function. After analysing the results, we conclude that the order of convergence is not related to the complexity of the probabilistic model, and that an algorithm whose probabilistic model mimics the structure of the objective function does not guarantee a low order of convergence