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Self-Adjusting Evolutionary Algorithms Are Slow on Multimodal Landscapes

Johannes Lengler, Konstantin Sturm

TL;DR

This paper analyzes self-adjusting evolutionary algorithms that use the one-fifth style rule to control the offspring population size in the SA-(1,λ)-EA. It evaluates performance on the distorted OneMax benchmark (disOM), where random distortions create local optima, and shows a lower runtime bound of $\Omega\left(\frac{n \ln n}{p}\right)$, indicating a slowdown by a factor $1/p$ relative to optimal static parameters. In contrast, a well-chosen static $\lambda$-EA achieves $O(n \ln n)$ and remains robust to distortion, a claim supported by empirical results. The findings suggest that self-adjusting approaches are not a universal remedy and motivate developing alternative adaptation strategies or portfolios that avoid the observed downsides on disOM.

Abstract

The one-fifth rule and its generalizations are a classical parameter control mechanism in discrete domains. They have also been transferred to control the offspring population size of the $(1, λ)$-EA. This has been shown to work very well for hill-climbing, and combined with a restart mechanism it was recently shown by Hevia Fajardo and Sudholt to improve performance on the multi-modal problem Cliff drastically. In this work we show that the positive results do not extend to other types of local optima. On the distorted OneMax benchmark, the self-adjusting $(1, λ)$-EA is slowed down just as elitist algorithms because self-adaptation prevents the algorithm from escaping from local optima. This makes the self-adaptive algorithm considerably worse than good static parameter choices, which do allow to escape from local optima efficiently. We show this theoretically and complement the result with empirical runtime results.

Self-Adjusting Evolutionary Algorithms Are Slow on Multimodal Landscapes

TL;DR

This paper analyzes self-adjusting evolutionary algorithms that use the one-fifth style rule to control the offspring population size in the SA-(1,λ)-EA. It evaluates performance on the distorted OneMax benchmark (disOM), where random distortions create local optima, and shows a lower runtime bound of , indicating a slowdown by a factor relative to optimal static parameters. In contrast, a well-chosen static -EA achieves and remains robust to distortion, a claim supported by empirical results. The findings suggest that self-adjusting approaches are not a universal remedy and motivate developing alternative adaptation strategies or portfolios that avoid the observed downsides on disOM.

Abstract

The one-fifth rule and its generalizations are a classical parameter control mechanism in discrete domains. They have also been transferred to control the offspring population size of the -EA. This has been shown to work very well for hill-climbing, and combined with a restart mechanism it was recently shown by Hevia Fajardo and Sudholt to improve performance on the multi-modal problem Cliff drastically. In this work we show that the positive results do not extend to other types of local optima. On the distorted OneMax benchmark, the self-adjusting -EA is slowed down just as elitist algorithms because self-adaptation prevents the algorithm from escaping from local optima. This makes the self-adaptive algorithm considerably worse than good static parameter choices, which do allow to escape from local optima efficiently. We show this theoretically and complement the result with empirical runtime results.
Paper Structure (11 sections, 9 theorems, 17 equations, 2 figures, 1 algorithm)

This paper contains 11 sections, 9 theorems, 17 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our Result
  3. Notation and Preliminaries
  4. General Notation.
  5. Distorted OneMax.
  6. Algorithms.
  7. Properties of the SA-${(1, \lambda)}$-EA
  8. Lower Runtime Bounds
  9. Experiments
  10. Conclusion
  11. Acknowledgements

Key Result

theorem thmcountertheorem

Consider the SA-${(1, \lambda)}$-EA with a resetting mechanism for the offspring population size on disOM with $p = \omega(1/n\ln{n})$, $d=\Omega(\ln{n})$, and $\lambda_{max} \ge n^{\Omega(1)}/p$. With high probability the algorithm takes $\Omega(n \ln{n}/p)$ function evaluations to reach a target f

Figures (2)

  • Figure 1: Normalized number of evaluations required by the SA-${(1, \lambda)}$-EA with resets to optimize disOM for different distortion probabilities $p$. We set $d= \ln{n}$, $k^\ast = n^{0.4}$, $F=1.5$, $s = 1$, $\lambda_{max} = n \ln{n}$ and average over $50$ runs each. The cutoff of $10^7$ evaluations was never reached. Note that the y-axis shows the number of evaluations $T$ multiplied by $p /(n \ln{n})$ and is scaled logarithmically.
  • Figure 2: We take the median over 50 runs for the $(1 , \lambda)$-EA, the $(1 + \lambda)$-EA and the SA-${(1, \lambda)}$-EA with resets. We set $d = \ln{n}$, $k^\ast = n^{0.4}$, $\lambda_{com, plus} = \lfloor 1.5 \ln{n}\rceil$ for the $(1 , \lambda)$-EA and the $(1 + \lambda)$-EA, $p = (e/(e-1))^{-\lambda_{com, plus}}$, $F= 1.5$, $s= 1$, $\lambda_{max} = n \ln{n}$. We make a cutoff at $10^{6}$ evaluations. We additionally plot the curve $n \ln{n}/p$ for reference.

Theorems & Definitions (18)

  • theorem thmcountertheorem
  • corollary thmcountercorollary
  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma
  • proof
  • ...and 8 more