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Fourier Analysis Meets Runtime Analysis: Precise Runtimes on Plateaus

Benjamin Doerr, Andrew James Kelley

TL;DR

This work introduces a novel discrete Fourier-analysis framework to analyze runtimes of time-evolutionary algorithms on plateaus, enabling exact hitting-time expressions previously unavailable for such symmetric landscapes. By modeling bit-flip mutations as a random walk on the abelian group $\mathbb{Z}_2^{\ell}$ and using Fourier representations, the authors derive precise runtimes for the $(1+1)$ EA on the Needle problem and a new Block-LeadingOnes benchmark. They obtain tight, asymptotically optimal mutation-rate strategies: a static rate $p = (1+o(1))\lambda/n$ with $\lambda$ minimizing $(e^x-1)/x^2$, and a fitness-dependent rate with $p_m \sim 1/(k+1)$ when $k$ bits are locked, plus explicit expressions for the corresponding runtimes. The results unify and extend LeadingOnes analyses to a broader plateau-class, showing the Fourier approach can yield exact constants and guide mutation-rate optimization, with potential applicability to other plateau problems and future population-based analyses.

Abstract

We propose a new method based on discrete Fourier analysis to analyze the time evolutionary algorithms spend on plateaus. This immediately gives a concise proof of the classic estimate of the expected runtime of the $(1+1)$ evolutionary algorithm on the Needle problem due to Garnier, Kallel, and Schoenauer (1999). We also use this method to analyze the runtime of the $(1+1)$ evolutionary algorithm on a new benchmark consisting of $n/\ell$ plateaus of effective size $2^\ell-1$ which have to be optimized sequentially in a LeadingOnes fashion. Using our new method, we determine the precise expected runtime both for static and fitness-dependent mutation rates. We also determine the asymptotically optimal static and fitness-dependent mutation rates. For $\ell = o(n)$, the optimal static mutation rate is approximately $1.59/n$. The optimal fitness dependent mutation rate, when the first $k$ fitness-relevant bits have been found, is asymptotically $1/(k+1)$. These results, so far only proven for the single-instance problem LeadingOnes, thus hold for a much broader class of problems. We expect similar extensions to be true for other important results on LeadingOnes. We are also optimistic that our Fourier analysis approach can be applied to other plateau problems as well.

Fourier Analysis Meets Runtime Analysis: Precise Runtimes on Plateaus

TL;DR

This work introduces a novel discrete Fourier-analysis framework to analyze runtimes of time-evolutionary algorithms on plateaus, enabling exact hitting-time expressions previously unavailable for such symmetric landscapes. By modeling bit-flip mutations as a random walk on the abelian group and using Fourier representations, the authors derive precise runtimes for the EA on the Needle problem and a new Block-LeadingOnes benchmark. They obtain tight, asymptotically optimal mutation-rate strategies: a static rate with minimizing , and a fitness-dependent rate with when bits are locked, plus explicit expressions for the corresponding runtimes. The results unify and extend LeadingOnes analyses to a broader plateau-class, showing the Fourier approach can yield exact constants and guide mutation-rate optimization, with potential applicability to other plateau problems and future population-based analyses.

Abstract

We propose a new method based on discrete Fourier analysis to analyze the time evolutionary algorithms spend on plateaus. This immediately gives a concise proof of the classic estimate of the expected runtime of the evolutionary algorithm on the Needle problem due to Garnier, Kallel, and Schoenauer (1999). We also use this method to analyze the runtime of the evolutionary algorithm on a new benchmark consisting of plateaus of effective size which have to be optimized sequentially in a LeadingOnes fashion. Using our new method, we determine the precise expected runtime both for static and fitness-dependent mutation rates. We also determine the asymptotically optimal static and fitness-dependent mutation rates. For , the optimal static mutation rate is approximately . The optimal fitness dependent mutation rate, when the first fitness-relevant bits have been found, is asymptotically . These results, so far only proven for the single-instance problem LeadingOnes, thus hold for a much broader class of problems. We expect similar extensions to be true for other important results on LeadingOnes. We are also optimistic that our Fourier analysis approach can be applied to other plateau problems as well.
Paper Structure (14 sections, 32 theorems, 154 equations)

This paper contains 14 sections, 32 theorems, 154 equations.

Table of Contents

  1. Introduction
  2. Previous Works
  3. Using Fourier Analysis
  4. Analysis of the Needle Problem
  5. The Block-LeadingOnes Problem
  6. Definition of the Block-LeadingOnes Problem
  7. Our Results
  8. Estimating the Runtime on One Block
  9. Approximating a Certain Sum
  10. The Optimal Static Mutation Rate
  11. Optimal Fitness-dependent Mutation Rate
  12. Erasing the $O(p)$ Term in Lemma \ref{['lem:simplified_expectation']}
  13. Proving Theorems \ref{['thm:optimal_mutation_rate']} and \ref{['thm:opt_adaptive_runtime']}
  14. Conclusion

Key Result

Theorem 3.1

Let $G$ be the abelian group $\mathbb{Z}_2^\ell$ with $2^\ell = m$, and let $\rho_0, \rho_1, \ldots, \rho_{m-1}$ be the characters of $G$, with $\rho_0$ being the trivial character $\rho_0 : G \to \{1\}$. Let $\mu$ be a probability distribution on $G$, and consider the random walk on $G$ generated b

Theorems & Definitions (61)

  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof
  • Proposition 3.4
  • proof
  • Corollary 4.1
  • Lemma 4.2
  • proof
  • ...and 51 more