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 the parameters μ and λ. If μ ≥ c log n for some constant c > 0 and λ = (1 + O(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 ≥ c'√n log n for a constant c' > 0 and λ = (1 + O(1))μ, the behavior of the algorithm changes and the bound on the expected runtime becomes O (μ√n), which typically even holds if the borders on the marginal probabilities are omitted. The results supplement the recently derived lower bound Ω (μ √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 p = c'√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).