Self-Adjusting Population Sizes for Non-Elitist Evolutionary Algorithms: Why Success Rates Matter

Recent theoretical studies have shown that self-adjusting mechanisms can provably outperform the best static parameters in evolutionary algorithms on discrete problems. However, the majority of these studies concerned elitist algorithms and we do not have a clear answer on whether the same mechanisms can be applied for non-elitist algorithms. We study one of the best-known parameter control mechanisms, the one-fifth success rule, to control the offspring population size $\lambda$ in the non-elitist ${(1 , \lambda)}$ EA. It is known that the ${(1 , \lambda)}$ EA has a sharp threshold with respect to the choice of $\lambda$ where the runtime on OneMax changes from polynomial to exponential time. Hence, it is not clear whether parameter control mechanisms are able to find and maintain suitable values of $\lambda$. We show that the answer crucially depends on the success rate $s$ (i.,e. a one-$(s+1)$-th success rule). We prove that, if the success rate is appropriately small, the self-adjusting ${(1 , \lambda)}$ EA optimises OneMax in $O(n)$ expected generations and $O(n \log n)$ expected evaluations. A small success rate is crucial: we also show that if the success rate is too large, the algorithm has an exponential runtime on OneMax.