Abstract
It is known that the \((1 + 1)\)-EA with mutation rate c/n optimises every monotone function efficiently if \(c<1\), and needs exponential time on some monotone functions (HotTopic functions) if \(c> c_0 = 2.13692..\). We study the same question for a large variety of algorithms, particularly for \((1 + \lambda )\)-EA, \((\mu + 1)\)-EA, \((\mu + 1)\)-GA, their fast counterparts like fast \((1 + 1)\)-EA, and for \((1 + (\lambda ,\lambda ))\)-GA. We prove that all considered mutation-based algorithms show a similar dichotomy for HotTopic functions, or even for all monotone functions. For the \((1 + (\lambda ,\lambda ))\)-GA, this dichotomy is in the parameter \(c\gamma \), which is the expected number of bit flips in an individual after mutation and crossover, neglecting selection. For the fast algorithms, the dichotomy is in \(m_2/m_1\), where \(m_1\) and \(m_2\) are the first and second falling moment of the number of bit flips. Surprisingly, the range of efficient parameters is not affected by either population size \(\mu \) nor by the offspring population size \(\lambda \).
The picture changes completely if crossover is allowed. The genetic algorithms \((\mu + 1)\)-GA and \((\mu + 1)\)-fGA are efficient for arbitrary mutations strengths if \(\mu \) is large enough.
Extended Abstract. All proofs and further details are available on arxiv [12].
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- 1.
We will be sloppy and drop the term “strictly” outside of theorems, but throughout the paper we always mean strictly monotone functions.
- 2.
Note that a heavy tail generally increases \(m_2\) much stronger than \(m_1\), so it increases the quotient \(m_2/m_1\).
- 3.
Note that this property might more correctly be called strictly monotone, but in this paper we will stick with the shorter, slightly less precise term monotone. In all other cases we use the standard terminology, e.g. the term increasing sequence has the same meaning as non-decreasing sequence.
- 4.
With high probability, i.e. with probability tending to one as \(n\rightarrow \infty \).
- 5.
In fact, the suggested parameter choice in [4, 6] satisfies \(c\gamma =1\) instead of \(c\gamma <1\). However, the runtime analysis in [6] only changes by constant factors if \(\gamma \) is decreased by a constant factor. Thus Theorem 2 applies to the parameter choices from [4, 6], except that \(\gamma \) is decreased by a constant factor.
- 6.
- 7.
This statement follows trivially from the other results by setting \(\mu =1\), and it is listed only for completeness.
- 8.
i.e., \(\Pr [\mathcal D = k] = k^{-\kappa }/\zeta (\kappa )\), where \(\zeta \) is the Riemann \(\zeta \) function.
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Lengler, J. (2018). A General Dichotomy of Evolutionary Algorithms on Monotone Functions. In: Auger, A., Fonseca, C., Lourenço, N., Machado, P., Paquete, L., Whitley, D. (eds) Parallel Problem Solving from Nature – PPSN XV. PPSN 2018. Lecture Notes in Computer Science(), vol 11102. Springer, Cham. https://doi.org/10.1007/978-3-319-99259-4_1
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