Runtime Analysis for Self-adaptive Mutation Rates

• Published in: Algorithmica Vol. 83; no. 4; pp. 1012 - 1053
• Main Authors: Doerr, Benjamin, Witt, Carsten, Yang, Jing
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.04.2021; Springer Nature B.V
• Subjects: 2018; Algorithm Analysis and Problem Complexity; Algorithms; Asymptotic properties; Computer Science; Computer Systems Organization and Communication Networks; Data Structures and Information Theory; Drift; Evolutionary algorithms; Evolutionary computation; Mathematics of Computing; Mutation; Optimization; Parameters; Run time (computers); Special Issue on Genetic and Evolutionary Computation; Theory of Computation; Two dimensional analysis; Runtime analysis; Self-adaptive; Evolutionary algorithms
• ISSN: 0178-4617; 1432-0541
• DOI: 10.1007/s00453-020-00726-2

We propose and analyze a self-adaptive version of the ( 1 , λ ) 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 λ / log λ + n log n ) when λ is at least C ln n for some constant C > 0 . For all values of λ ≥ C ln n , this performance is asymptotically best possible among all λ -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, Gießen, 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.