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A Runtime Analysis of the Multi-Valued Compact Genetic Algorithm on Generalized LeadingOnes

Sumit Adak, Carsten Witt

TL;DR

The paper analyzes the runtime of the multi-valued compact genetic algorithm ($r$-cGA) on the $r$-valued LeadingOnes problem, establishing a high-probability runtime bound of $O(n^2 r^2 \log^3 n \log^2 r)$ under suitable parameter choices. By modeling the marginals with a frequency matrix, classifying updates into random-walk and biased steps, and employing border restrictions to prevent genetic drift, the authors develop a two-stage analysis combining additive drift with tail bounds and multiplicative drift for the critical phases. The work extends existing runtime results for univariate and multi-valued EDAs, providing the first runtime analysis of the binary and multi-valued cGA on LeadingOnes and connecting to recent multi-valued EDAs on related problems. Empirical results corroborate border-effect considerations and suggest room for tightening the theoretical bounds, while reinforcing the practical viability of r-valued EDAs on structured problems. Overall, the study contributes a rigorous framework for understanding how univariate, multi-valued EDAs behave on a canonical, positionally structured objective, with implications for parameter selection and drift control in more general multi-valued optimization tasks.

Abstract

In the literature on runtime analyses of estimation of distribution algorithms (EDAs), researchers have recently explored univariate EDAs for multi-valued decision variables. Particularly, Jedidia et al. gave the first runtime analysis of the multi-valued UMDA on the r-valued LeadingOnes (r-LeadingOnes) functions and Adak et al. gave the first runtime analysis of the multi-valued cGA (r-cGA) on the r-valued OneMax function. We utilize their framework to conduct an analysis of the multi-valued cGA on the r-valued LeadingOnes function. Even for the binary case, a runtime analysis of the classical cGA on LeadingOnes was not yet available. In this work, we show that the runtime of the r-cGA on r-LeadingOnes is O(n^2r^2 log^3 n log^2 r) with high probability.

A Runtime Analysis of the Multi-Valued Compact Genetic Algorithm on Generalized LeadingOnes

TL;DR

The paper analyzes the runtime of the multi-valued compact genetic algorithm (-cGA) on the -valued LeadingOnes problem, establishing a high-probability runtime bound of under suitable parameter choices. By modeling the marginals with a frequency matrix, classifying updates into random-walk and biased steps, and employing border restrictions to prevent genetic drift, the authors develop a two-stage analysis combining additive drift with tail bounds and multiplicative drift for the critical phases. The work extends existing runtime results for univariate and multi-valued EDAs, providing the first runtime analysis of the binary and multi-valued cGA on LeadingOnes and connecting to recent multi-valued EDAs on related problems. Empirical results corroborate border-effect considerations and suggest room for tightening the theoretical bounds, while reinforcing the practical viability of r-valued EDAs on structured problems. Overall, the study contributes a rigorous framework for understanding how univariate, multi-valued EDAs behave on a canonical, positionally structured objective, with implications for parameter selection and drift control in more general multi-valued optimization tasks.

Abstract

In the literature on runtime analyses of estimation of distribution algorithms (EDAs), researchers have recently explored univariate EDAs for multi-valued decision variables. Particularly, Jedidia et al. gave the first runtime analysis of the multi-valued UMDA on the r-valued LeadingOnes (r-LeadingOnes) functions and Adak et al. gave the first runtime analysis of the multi-valued cGA (r-cGA) on the r-valued OneMax function. We utilize their framework to conduct an analysis of the multi-valued cGA on the r-valued LeadingOnes function. Even for the binary case, a runtime analysis of the classical cGA on LeadingOnes was not yet available. In this work, we show that the runtime of the r-cGA on r-LeadingOnes is O(n^2r^2 log^3 n log^2 r) with high probability.
Paper Structure (12 sections, 8 theorems, 23 equations, 2 figures, 1 algorithm)

This paper contains 12 sections, 8 theorems, 23 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Preliminaries
  4. Related Work
  5. The Multi-valued cGA
  6. Genetic Drift
  7. Behavior of the Probabilistic Model
  8. Analysis of Genetic Drift for the $r$-cGA
  9. Runtime Analysis
  10. Experiments
  11. Conclusion
  12. Acknowledgments.

Key Result

theorem thmcountertheorem

Let $a_1,\dots,a_m\in \mathbb{R}$, and $X_1,\dots,X_m$ be a martingale difference sequence with $\lvert X_k \rvert\leq a_k$ for each $k$. Then for all $\varepsilon \in \mathbb{R}_{\geq 0}$, it holds that

Figures (2)

  • Figure 1: Empirical runtime of the $r$-cGA on $r$-LeadingOnes; for $n=500$ (left-hand side) and $n=1000$ (right-hand side), $K\in\{100,\dots,1000\}$ and averaged over 1000 runs.
  • Figure 2: Empirical runtime of the $r$-cGA on $r$-LeadingOnes; for $r=2$ (top-left), $r=3$ (top-middle), $r=4$ (top-right), $r=5$ (bottom-left), $r=6$ (bottom-middle) and $r=7$ (bottom-right); $n\in\{100,\dots,400\}$, $K\in\{100,\dots,1000\}$ and averaged over 200 runs.

Theorems & Definitions (11)

  • theorem thmcountertheorem
  • theorem thmcountertheorem
  • proof
  • theorem thmcountertheorem
  • proof
  • theorem thmcountertheorem
  • lemma thmcounterlemma
  • corollary thmcountercorollary
  • lemma thmcounterlemma: Theorem 7 in KLWFOGA15
  • lemma thmcounterlemma
  • ...and 1 more