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Does Comma Selection Help to Cope with Local Optima?

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Abstract

One hope when using non-elitism in evolutionary computation is that the ability to abandon the current-best solution aids leaving local optima. To improve our understanding of this mechanism, we perform a rigorous runtime analysis of a basic non-elitist evolutionary algorithm (EA), the \((\mu ,\lambda )\) EA, on the most basic benchmark function with a local optimum, the jump function. We prove that for all reasonable values of the parameters and the problem, the expected runtime of the \((\mu ,\lambda )\) EA is, apart from lower order terms, at least as large as the expected runtime of its elitist counterpart, the \((\mu +\lambda )\) EA (for which we conduct the first runtime analysis on jump functions to allow this comparison). Consequently, the ability of the \((\mu ,\lambda )\) EA to leave local optima to inferior solutions does not lead to a runtime advantage. We complement this lower bound with an upper bound that, for broad ranges of the parameters, is identical to our lower bound apart from lower order terms. This is the first runtime result for a non-elitist algorithm on a multi-modal problem that is tight apart from lower order terms.

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Notes

  1. We use the term precise to denote runtime estimates that are asymptotically tight including the leading constant, that is, where the estimated runtime \({\tilde{T}}(n)\) and the true runtime T(n) for problem size n satisfy \(\lim \limits _{n \rightarrow \infty } {\tilde{T}}(n) / T(n) = 1\).

  2. To ease the presentation, we only consider the standard mutation rate 1/n, but we are confident that our results in an analogous fashion hold for general mutation rates \(\chi /n\), \(\chi \) a constant. Previous works have shown that the constant \(\chi \) has an influence (again by constant factors) on where the boundary between the “imitating elitism” and “no efficient progress” regimes is located. Since our result is that the \({(\mu ,\lambda )}~\mathrm{EA}\) for no realistic parameter settings beats the \((\mu +\lambda )~\mathrm{EA}\), we do not expect that a constant factor change of the mutation rate leads to substantially different findings.

  3. More precisely, Cn could be replaced by \(t_0\) from (6).

  4. This argument, ignoring however the initial search points, was made already in [15] to show this runtime bound for the \((1 + 1)~\mathrm{EA}\).

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Acknowledgements

This work was supported by a public grant as part of the Investissements d’avenir project, reference ANR-11-LABX-0056-LMH, LabEx LMH.

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

    Benjamin Doerr

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This is the full version of a paper [27] that appeared at GECCO 2020. It contains all proofs and additional information that had to be omitted from the conference version for reasons of space.

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Doerr, B. Does Comma Selection Help to Cope with Local Optima?. Algorithmica 84, 1659–1693 (2022). https://doi.org/10.1007/s00453-021-00896-7

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