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A Rigorous Runtime Analysis of the \((1 + (\lambda , \lambda ))\) GA on Jump Functions

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Abstract

The \((1 + (\lambda ,\lambda ))\) genetic algorithm is a younger evolutionary algorithm trying to profit also from inferior solutions. Rigorous runtime analyses on unimodal fitness functions showed that it can indeed be faster than classical evolutionary algorithms, though on these simple problems the gains were only moderate. In this work, we conduct the first runtime analysis of this algorithm on a multimodal problem class, the jump functions benchmark. We show that with the right parameters, the \({(1 + (\lambda , \lambda ))}\) GA optimizes any jump function with jump size \(2 \le k \le n/4\) in expected time \(O(n^{(k+1)/2} e^{O(k)} k^{-k/2})\), which significantly and already for constant k outperforms standard mutation-based algorithms with their \(\Theta (n^k)\) runtime and standard crossover-based algorithms with their \({\tilde{O}}(n^{k-1})\) runtime guarantee. For the isolated problem of leaving the local optimum of jump functions, we determine provably optimal parameters that lead to a runtime of \((n/k)^{k/2} e^{\Theta (k)}\). This suggests some general advice on how to set the parameters of the \({(1 + (\lambda , \lambda ))}\) GA, which might ease the further use of this algorithm.

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Notes

  1. This particular structure of the jump benchmark has been criticized and several variants have been proposed [8, 37, 46]. With the overwhelming majority of the runtime analyses on multimodal problems still regarding the classic jump benchmark, for the sake of comparability we prefer to regard this benchmark as well.

  2. In [34] it was shown that the \({(1 + (\lambda , \lambda ))}\) GA with the standard parameter setting is not effective on Jump even when parameter control mechanisms are applied to \(\lambda \). This is another reason to step back from the standard parameters and consider a wider parameters space.

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Acknowledgements

This work was supported by a public Grant as part of the Investissement d’avenir project, reference ANR-11-LABX-0056-LMH, LabEx LMH and by RFBR and CNRS, Project No. 20-51-15009.

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  1. ITMO University, St. Petersburg, Russia

    Denis Antipov & Vitalii Karavaev

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

    Benjamin Doerr

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  1. Denis Antipov
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Extended version of the paper [6] in the proceedings of GECCO 2020. This version contains all proofs and other details that had to be omitted in the conference version for reasons of space. Also, we have added a new section which proves the lower bounds.

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Antipov, D., Doerr, B. & Karavaev, V. A Rigorous Runtime Analysis of the \((1 + (\lambda , \lambda ))\) GA on Jump Functions. Algorithmica 84, 1573–1602 (2022). https://doi.org/10.1007/s00453-021-00907-7

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