Abstract
We study the \((1:s+1)\) success rule for controlling the population size of the \((1,\lambda )\)-EA. It was shown by Hevia Fajardo and Sudholt that this parameter control mechanism can run into problems for large s if the fitness landscape is too easy. They conjectured that this problem is worst for the OneMax benchmark, since in some well-established sense OneMax is known to be the easiest fitness landscape.
In this paper we disprove this conjecture. We show that there exist s and \(\varepsilon \) such that the self-adjusting \((1,\lambda )\)-EA with the \((1:s+1)\)-rule optimizes OneMax efficiently when started with \(\varepsilon n\) zero-bits, but does not find the optimum in polynomial time on Dynamic BinVal. Hence, we show that there are landscapes where the problem of the \((1:s+1)\)-rule for controlling the population size of the \((1 , \lambda )\)-EA is more severe than for OneMax. The key insight is that, while OneMax is the easiest function for decreasing the distance to the optimum, it is not the easiest fitness landscape with respect to finding fitness-improving steps.
M. Kaufmann—The author was supported by the Swiss National Science Foundation [grant number 200021_192079].
X. Zou—The author was supported by the Swiss National Science Foundation [grant number CR-SII5_173721].
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Notes
- 1.
Traditionally the most popular value is \(s=4\), leading to the famous one-fifth rule [2].
- 2.
Empirically they found the threshold between the two regimes to be around \(s\approx 3.4\).
- 3.
See [23, Section 2.1] for a detailed discussion of the target population size \(\lambda ^*\).
- 4.
A function \(f:\{0,1\}^n \rightarrow \mathbb {R}\) is monotone if flipping a zero-bit into a one-bit always improves the fitness. In the dynamic monotone setting, selection may be based on a different function in each generation, but it must always be a monotone function. The formal definition is not relevant for this paper, but can be found in [23].
- 5.
In Lemma 7 () and Lemma 13 we consider a constant \(\varepsilon >0\), introduce an \(\varepsilon ' = \varepsilon '(\varepsilon )\) and look at all states in the range \((\varepsilon \pm \varepsilon ')n\). Again, we implicitly assume that \((\varepsilon - \varepsilon ')n\) and \((\varepsilon + \varepsilon ')n\) are integers, since we use those in the calculations.
- 6.
The subscript indicates dependency on \(\lambda \), i.e., for all \(c,C >0\) there exists \(\lambda _0\) such that for all \(\lambda \ge \lambda _0\) we have \(\tilde{\varepsilon }(\lambda ) \le c/\lambda \) and \(\tilde{s}(\lambda ) \ge C\).
- 7.
Defined as \(\textsc {BinVal} (x) = \sum _{i=1}^{n} 2^{i-1} x_i \) and \(\textsc {Binary} (x) = \sum _{i=1}^{\lfloor n/2 \rfloor } x_i n + \sum _{i=\lfloor n/2 \rfloor + 1}^n x_i\).
- 8.
We note that there are other types of “easiness”, e.g. with respect to a fixed budget.
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Acknowledgements
We thank Dirk Sudholt for helpful discussions during the Dagstuhl seminar 22081 “Theory of Randomized Optimization Heuristics” and Mario Hevia Fajardo for sharing his simulation code for comparison.
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Kaufmann, M., Larcher, M., Lengler, J., Zou, X. (2023). OneMax Is Not the Easiest Function for Fitness Improvements. In: Pérez Cáceres, L., Stützle, T. (eds) Evolutionary Computation in Combinatorial Optimization. EvoCOP 2023. Lecture Notes in Computer Science, vol 13987. Springer, Cham. https://doi.org/10.1007/978-3-031-30035-6_11
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