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,$λ$) EA combined with a self-adjusting offspring population size $λ$ 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,$λ$) 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,$λ$) 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.