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Static and Self-Adjusting Mutation Strengths for Multi-valued Decision Variables

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Abstract

The most common representation in evolutionary computation are bit strings. With very little theoretical work existing on how to use evolutionary algorithms for decision variables taking more than two values, we study the run time of simple evolutionary algorithms on some OneMax-like functions defined over \(\varOmega = \{0, 1, \ldots , r-1\}^n\). We observe a crucial difference in how we extend the one-bit-flip and standard-bit mutation operators to the multi-valued domain. While it is natural to modify a random position of the string or select each position of the solution vector for modification independently with probability 1/n, there are various ways to then change such a position. If we change each selected position to a random value different from the original one, we obtain an expected run time of \(\varTheta (nr \log n)\). If we change each selected position by \(+1\) or \(-1\) (random choice), the optimization time reduces to \(\varTheta (nr + n\log n)\). If we use a random mutation strength \(i \in \{0,1,\ldots ,r-1\}\) with probability inversely proportional to i and change the selected position by \(+i\) or \(-i\) (random choice), then the optimization time becomes \(\varTheta (n \log (r)(\log n +\log r))\), which is asymptotically faster than the previous if \(r = \omega (\log (n) \log \log (n))\). Interestingly, a better expected performance can be achieved with a self-adjusting mutation strength that is based on the success of previous iterations. For the mutation operator that modifies a randomly chosen position, we show that the self-adjusting mutation strength yields an expected optimization time of \(\varTheta (n (\log n + \log r))\), which is best possible among all dynamic mutation strengths. In our proofs, we use a new multiplicative drift theorem for computing lower bounds, which is not restricted to processes that move only towards the target.

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Notes

  1. Following the terminology introduced in [24] and extended in [10, Section 3.1] we distinguish adaptive parameter choices into functionally-dependent and self-adjusting ones. While functionally-dependent parameter choices depend only on the current state of the algorithm, they may explicitly use absolute fitness values. Fitness- and rank-dependent mutation rates are a typical example for such functionally-dependent parameter choices. Self-adjusting parameter choices, in contrast, do not depend on absolute fitness information but rather on the success of previous iterations. This is the case of the parameter updates of the \(\mathtt{RLS} _{a,b}\) considered in this work. A typical representative of this class is the so-called one-fifth rule that is often used in evolution strategies for controlling the step size of the algorithm under consideration. Other dynamic update rules are either called deterministic—this is the case if there is no dependency between the parameters and the success or state of the optimization process other than the iteration count—or self-adapting. Self-adaptive algorithms code the parameters themselves into the genome of the individuals and hope to evolve good parameters during the optimization process.

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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, in a joint call with Programme Gaspard Monge en Optimisation et Recherche Opérationnelle.

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Authors and Affiliations

  1. LIX - UMR 7161, École Polytechnique, CS35003, 91120, Palaiseau, France

    Benjamin Doerr

  2. CNRS, LIP6 UMR 7606, Sorbonne Universités, UPMC Univ Paris 06, 4 place Jussieu, 75005, Paris, France

    Carola Doerr

  3. Hasso-Plattner-Institut, Prof.-Dr.-Helmert-Str. 2-3, 14482, Potsdam, Germany

    Timo Kötzing

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  1. Benjamin Doerr
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Correspondence to Timo Kötzing.

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Results presented in this work are based on [12, 13].

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Doerr, B., Doerr, C. & Kötzing, T. Static and Self-Adjusting Mutation Strengths for Multi-valued Decision Variables. Algorithmica 80, 1732–1768 (2018). https://doi.org/10.1007/s00453-017-0341-1

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