Hard Problems are Easier for Success-based Parameter Control

Recent works showed that simple success-based rules for self-adjusting parameters in evolutionary algorithms (EAs) can match or outperform the best fixed parameters on discrete problems. Non-elitism in a (1,$\lambda$) EA combined with a self-adjusting offspring population size $\lambda$ outperforms common EAs on the multimodal Cliff problem. However, it was shown that this only holds if the success rate $s$ that governs self-adjustment is small enough. Otherwise, even on OneMax, the self-adjusting (1,$\lambda$) EA stagnates on an easy slope, where frequent successes drive down the offspring population size. We show that self-adjustment works as intended in the absence of easy slopes. We define everywhere hard functions, for which successes are never easy to find and show that the self-adjusting (1,$\lambda$) EA is robust with respect to the choice of success rates $s$. We give a general fitness-level upper bound on the number of evaluations and show that the expected number of generations is at most $O(d + \log(1/p_{\min}))$ where $d$ is the number of non-optimal fitness values and $p_{\min}$ is the smallest probability of finding an improvement from a non-optimal search point. We discuss implications for the everywhere hard function LeadingOnes and a new class OneMaxBlocks of everywhere hard functions with tunable difficulty.

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.