Parallel Black-Box Complexity with Tail Bounds

We propose a new black-box complexity model for search algorithms evaluating $λ$ search points in parallel. The parallel unary unbiased black-box complexity gives lower bounds on the number of function evaluations every parallel unary unbiased black-box algorithm needs to optimise a given problem. It captures the inertia caused by offspring populations in evolutionary algorithms and the total computational effort in parallel metaheuristics. We present complexity results for LeadingOnes and OneMax. Our main result is a general performance limit: we prove that on every function every $λ$-parallel unary unbiased algorithm needs at least $Ω(\frac{λn}{\ln λ} + n \log n)$ evaluations to find any desired target set of up to exponential size, with an overwhelming probability. This yields lower bounds for the typical optimisation time on unimodal and multimodal problems, for the time to find any local optimum, and for the time to even get close to any optimum. The power and versatility of this approach is shown for a wide range of illustrative problems from combinatorial optimisation. Our performance limits can guide parameter choice and algorithm design; we demonstrate the latter by presenting an optimal $λ$-parallel algorithm for OneMax that uses parallelism most effectively.