Runtime Analysis for Self-adaptive Mutation Rates

• Main Authors: Doerr, Benjamin, Witt, Carsten, Yang, Jing
• Format: Journal Article
• Language: English
• Published: 30.11.2018
• Subjects: Computer Science - Artificial Intelligence; Computer Science - Data Structures and Algorithms; Computer Science - Learning; Computer Science - Neural and Evolutionary Computing
• DOI: 10.48550/arxiv.1811.12824

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.