Core Concepts
We provide a tight upper bound of O(μn log(k) + 4k/pc) for the runtime of a variant of the (μ+1) Genetic Algorithm on the Jump_k benchmark, under mild assumptions on the population size μ and crossover probability pc.
Abstract
The authors analyze the evolution of population diversity, measured as the sum of pairwise Hamming distances, for a variant of the (μ+1) Genetic Algorithm (GA) on the Jump_k benchmark. They show that the population diversity converges to an equilibrium of near-perfect diversity. This allows them to derive an improved and tight time bound of O(μn log(k) + 4k/pc) for a range of k, under the mild assumptions pc = O(1/k) and μ ∈ Ω(kn).
The key insights are:
For large crossover probabilities pc, the authors show that a constant fraction of all crossovers happens between parents of Hamming distance 2k, which is the largest possible Hamming distance on the set of local optima (plateau) of Jump_k.
The authors build on their previous work on equilibrium states for population diversity on flat fitness functions and translate this approach to the Jump_k plateau.
The runtime bound improves upon previous results by allowing larger crossover probabilities pc = O(1/k) compared to the previous bound of pc = O(1/(kn)), and it holds for a larger range of gap lengths k = o(√n) as opposed to k = O(log n).
For constant k, the authors show that the (μ+1) GA with competing crossover offspring takes expected time O(μn log μ), a massive improvement over the previously best result of O(nk-1) in this range.