Runtime Analysis for Self-adaptive Mutation Rates

Abstract

We propose and analyze a self-adaptive version of the $$(1,\lambda )$$ evolutionary algorithm in which the current mutation rate is encoded within the individual and thus also subject to mutation. A rigorous runtime analysis on the OneMax benchmark function reveals that a simple local mutation scheme for the rate leads to an expected optimization time (number of fitness evaluations) of $$O(n\lambda /\log \lambda +n\log n)$$ when $$\lambda$$ is at least $$C \ln n$$ for some constant $$C > 0$$ . For all values of $$\lambda \ge C \ln n$$ , this performance is asymptotically best possible among all $$\lambda$$ -parallel mutation-based unbiased black-box algorithms. Our result rigorously proves for the first time that self-adaptation in evolutionary computation can find complex optimal parameter settings on the fly. In particular, it gives asymptotically the same performance as the relatively complicated self-adjusting scheme for the mutation rate proposed by Doerr, Giesen, Witt, and Yang (Algorithmica 2019). On the technical side, the paper contributes new tools for the analysis of two-dimensional drift processes arising in the analysis of dynamic parameter choices in EAs, including bounds on occupation probabilities in processes with non-constant drift.