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Runtime Analysis for Self-adaptive Mutation Rates

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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, 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.

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Notes

  1. In this part of the proof, we use the fact that \(F=32\). This does not mean that for other not too small values of F we would not obtain similar results, but it increases the readability to work with this concrete value.

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Acknowledgements

The authors thank Christian Gießen for useful discussions on this topic. This work was supported by a public grant as part of the Investissement d’avenir project, reference ANR-11-LABX-0056-LMH, LabEx LMH, in a joint call with Gaspard Monge Program for optimization, operations research and their interactions with data sciences. This publication is based upon work from COST Action CA15140, supported by COST.

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  1. Laboratoire d’Informatique (LIX), CNRS, École Polytechnique, Institut Polytechnique de Paris, Palaiseau, France

    Benjamin Doerr & Jing Yang

  2. DTU Compute, Technical University of Denmark, Kongens Lyngby, Denmark

    Carsten Witt

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  1. Benjamin Doerr
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Correspondence to Carsten Witt.

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Extended version of a paper appearing at the Genetic and Evolutionary Computation Conference 2018 [35]. This version contains all proofs, whereas most of them for reasons of space did not fit into the conference version. In this version, the main result is valid for all \(\lambda \ge C \ln (n)\), C a sufficiently large constant, whereas the conference version needed \(\lambda \ge (\ln n)^{1+\varepsilon }\) for an arbitrary \(\varepsilon >0\).

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Doerr, B., Witt, C. & Yang, J. Runtime Analysis for Self-adaptive Mutation Rates. Algorithmica 83, 1012–1053 (2021). https://doi.org/10.1007/s00453-020-00726-2

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