Runtime Analysis for Self-adaptive Mutation Rates - Doerr, Benjamin; Witt, Carsten; Yang, Jing (2018)
We propose and analyze a self-adaptive version of the$(1,\lambda)$evolutionary algorithm in which the current mutation rate is part of 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 shows that self-adaptation in evolutionary computation can find complex optimal parameter settings on the fly. At the same time, it proves that a relatively complicated self-adjusting scheme for the mutation rate proposed by Doerr, Gießen, Witt, and Yang~(GECCO~2017) can be replaced by our simple endogenous scheme. 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.