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Lower Bounds for Comparison Based Evolution Strategies Using VC-dimension and Sign Patterns

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Abstract

We derive lower bounds on the convergence rate of comparison based or selection based algorithms, improving existing results in the continuous setting, and extending them to non-trivial results in the discrete case. This is achieved by considering the VC-dimension of the level sets of the fitness functions; results are then obtained through the use of the shatter function lemma. In the special case of optimization of the sphere function, improved lower bounds are obtained by an argument based on the number of sign patterns.

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Authors and Affiliations

  1. Laboratoire PRiSM, CNRS UMR 8144 and Univ. Versailles Saint-Quentin-en-Yvelines, 45 av. des États-Unis, 78035, Versailles, France

    Hervé Fournier

  2. TAO (Inria), LRI, UMR 8623 (CNRS-Univ. Paris-Sud), Bât 490, Univ. Paris-Sud, 91405, Orsay, France

    Olivier Teytaud

Authors
  1. Hervé Fournier
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  2. Olivier Teytaud
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Correspondence to Hervé Fournier.

Additional information

A preliminary version of this paper appeared in PPSN 2008 [27].

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Fournier, H., Teytaud, O. Lower Bounds for Comparison Based Evolution Strategies Using VC-dimension and Sign Patterns. Algorithmica 59, 387–408 (2011). https://doi.org/10.1007/s00453-010-9391-3

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