Black-box complexity of parallel search with distributed populations

Many metaheuristics such as island models and cellular evolutionary algorithms use a network of distributed populations that communicate search points along a spatial communication topology. The idea is to slow down the spread of information, reducing the risk of "premature convergence", and sacrificing exploitation for an increased exploration. We introduce the distributed black-box complexity as the minimum number of function evaluations every distributed black-box algorithm needs to optimise a given problem. It depends on the topology, the number of populations λ, and the problem size n. We give upper and lower bounds on the distributed black-box complexity for unary unbiased blackbox algorithms on a class of unimodal functions in order to study the impact of communication topologies on performance. Our results confirm that rings and torus graphs can lead to higher black-box complexities, compared to unrestricted communication. We further determine cut-off points for the number of populations λ, above which no distributed black-box algorithm can achieve linear speedups.