Convergence Analysis of Evolution Strategies with Random Numbers of Offspring

Abstract. Hitting times of the global optimum for evolutionary algorithms are usually available for simple unimodal problems or for simplied algorithms. In discrete problems, the number of results that relate the convergence rate of evolution strategies to the geometry of the optimisation landscape is restricted to a few theoretical studies. This article introduces a variant of the canonical (μ+ λ)-ES, called the Poisson-ES, for which the number of o spring is not deterministic, but is instead sampled from a Poisson distribution with mean λ. After a slight change on the rank-based selection for the μ parents, and assuming that the number of o spring is small, we show that the convergence rate of the new algorithm is dependent on a geometric quantity that measures the maximal width of adaptive valleys. The argument of the proof is based on the analogy of the Poisson-ES with a basic Mutation-or-Selection evolutionary strategy introduced in a previous work.