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Erschienen in:
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2018 | OriginalPaper | Buchkapitel

A General Dichotomy of Evolutionary Algorithms on Monotone Functions

verfasst von : Johannes Lengler

Erschienen in: Parallel Problem Solving from Nature – PPSN XV

Verlag: Springer International Publishing

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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.

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Fußnoten
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
Note that this is not a trivial consequence of Theorem 4, since  (6), (7) are conditions on the distribution for the best of \(\lambda \) offspring, while the condition here is on the distribution \(\mathcal D\) for generating a single offspring.
 
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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Metadaten
Titel
A General Dichotomy of Evolutionary Algorithms on Monotone Functions
verfasst von
Johannes Lengler
Copyright-Jahr
2018
Buch
Parallel Problem Solving from Nature – PPSN XV

Print ISBN: 978-3-319-99258-7

Electronic ISBN: 978-3-319-99259-4

Copyright-Jahr: 2018

DOI
https://doi.org/10.1007/978-3-319-99259-4_1

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