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The \((1+1)\) Elitist Black-Box Complexity of LeadingOnes

Abstract

One important goal of black-box complexity theory is the development of complexity models allowing to derive meaningful lower bounds for whole classes of randomized search heuristics. Complementing classical runtime analysis, black-box models help us to understand how algorithmic choices such as the population size, the variation operators, or the selection rules influence the optimization time. One example for such a result is the \(\varOmega (n \log n)\) lower bound for unary unbiased algorithms on functions with a unique global optimum (Lehre and Witt in Algorithmica 64:623–642, 2012), which tells us that higher arity operators or biased sampling strategies are needed when trying to beat this bound. In lack of analyzing techniques, such non-trivial lower bounds are very rare in the existing literature on black-box optimization and therefore remain to be one of the main challenges in black-box complexity theory. With this paper we contribute to our technical toolbox for lower bound computations by proposing a new type of information-theoretic argument. We regard the permutation- and bit-invariant version of LeadingOnes and prove that its \((1+1)\) elitist black-box complexity is \(\varOmega (n^2)\), a bound that is matched by \((1+1)\)-type evolutionary algorithms. The \((1+1)\) elitist complexity of LeadingOnes is thus considerably larger than its unrestricted one, which is known to be of order \(n\log \log n\) (Afshani et al. in Lecture notes in computer science, vol 8066, pp 1–11. Springer, New York, 2013). The \(\varOmega (n^2)\) lower bound does not rely on the fact that elitist black-box algorithms are not allowed to make use of absolute fitness values. In contrast, we show that even if absolute fitness values are revealed to the otherwise elitist algorithm, it cannot significantly profit from this additional information. Our result thus shows that for LeadingOnes the memory-restriction, together with the selection requirement, has a substantial impact on the best possible performance.

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Notes

  1. We remark without proof that the same argument also works in a different model of computation where the algorithm is allowed to generate random real number in the interval [0, 1].

  2. For every i we fix \(n_i\) arbitrarily for which the statement holds when \(p=c=1/i\). Without loss of generality, we may assume that the \(n_i\) are growing (by assumption, for every i there are arbitrary large values for n for which some algorithm A spends at least time cn with probability at most p). Then we maychoose the functions \(p(n), c(n) = o(1)\) so slowly decreasing that \(p(n_i),c(n_i) \ge 1/i\). For these functions p(n), c(n), there are still arbitrarily large n for which the statement holds, since it holds for all triples \((n_i, p(n_i), c(n_i))\).

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Acknowledgements

This research benefited from the support of the “FMJH Program Gaspard Monge in optimization and operation research”, and from the support to this program from EDF (Électricité de France).

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Correspondence to Johannes Lengler.

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Doerr, C., Lengler, J. The \((1+1)\) Elitist Black-Box Complexity of LeadingOnes. Algorithmica 80, 1579–1603 (2018). https://doi.org/10.1007/s00453-017-0304-6

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Keywords

  • Black-box complexity
  • Query complexity
  • LeadingOnes
  • Elitist algorithm
  • Memory restriction
  • Truncation selection
  • Evolutionary algorithms