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Improved Runtime Analysis of a Multi-Valued Compact Genetic Algorithm on Two Generalized OneMax Problems

Sumit Adak, Carsten Witt

TL;DR

This work extends runtime analysis of univariate EDAs to multi-valued settings by addressing the $r$-cGA on generalized OneMax problems. It proves a first theoretical bound of $O(n r^3 \log^2 n \log r)$ for G-OneMax and improves the $r$-OneMax bound to $O(n r \log n \log r)$, while incorporating frequency borders into the analysis. The authors develop a negative-drift framework to control genetic drift and employ additive drift with tail bounds across frequency-phase transitions, both with and without borders, and validate findings with experimentation. These results broaden the understanding of EDAs on multi-valued spaces and suggest potential for further runtime improvements in complex problem settings.

Abstract

Recent research in the runtime analysis of estimation of distribution algorithms (EDAs) has focused on univariate EDAs for multi-valued decision variables. In particular, the runtime of the multi-valued cGA (r-cGA) and UMDA on multi-valued functions has been a significant area of study. Adak and Witt (PPSN 2024) and Hamano et al. (ECJ 2024) independently performed a first runtime analysis of the r-cGA on the r-valued OneMax function (r-OneMax). Adak and Witt also introduced a different r-valued OneMax function called G-OneMax. However, for that function, only empirical results were provided so far due to the increased complexity of its runtime analysis, since r-OneMax involves categorical values of two types only, while G-OneMax encompasses all possible values. In this paper, we present the first theoretical runtime analysis of the r-cGA on the G-OneMax function. We demonstrate that the runtime is O(nr^3 log^2 n log r) with high probability. Additionally, we refine the previously established runtime analysis of the r-cGA on r-OneMax, improving the previous bound to O(nr log n log r), which improves the state of the art by an asymptotic factor of log n and is tight for the binary case. Moreover, we for the first time include the case of frequency borders.

Improved Runtime Analysis of a Multi-Valued Compact Genetic Algorithm on Two Generalized OneMax Problems

TL;DR

This work extends runtime analysis of univariate EDAs to multi-valued settings by addressing the -cGA on generalized OneMax problems. It proves a first theoretical bound of for G-OneMax and improves the -OneMax bound to , while incorporating frequency borders into the analysis. The authors develop a negative-drift framework to control genetic drift and employ additive drift with tail bounds across frequency-phase transitions, both with and without borders, and validate findings with experimentation. These results broaden the understanding of EDAs on multi-valued spaces and suggest potential for further runtime improvements in complex problem settings.

Abstract

Recent research in the runtime analysis of estimation of distribution algorithms (EDAs) has focused on univariate EDAs for multi-valued decision variables. In particular, the runtime of the multi-valued cGA (r-cGA) and UMDA on multi-valued functions has been a significant area of study. Adak and Witt (PPSN 2024) and Hamano et al. (ECJ 2024) independently performed a first runtime analysis of the r-cGA on the r-valued OneMax function (r-OneMax). Adak and Witt also introduced a different r-valued OneMax function called G-OneMax. However, for that function, only empirical results were provided so far due to the increased complexity of its runtime analysis, since r-OneMax involves categorical values of two types only, while G-OneMax encompasses all possible values. In this paper, we present the first theoretical runtime analysis of the r-cGA on the G-OneMax function. We demonstrate that the runtime is O(nr^3 log^2 n log r) with high probability. Additionally, we refine the previously established runtime analysis of the r-cGA on r-OneMax, improving the previous bound to O(nr log n log r), which improves the state of the art by an asymptotic factor of log n and is tight for the binary case. Moreover, we for the first time include the case of frequency borders.

Paper Structure

This paper contains 11 sections, 12 theorems, 13 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Framework for the Analysis
  4. Analysis of Runtime
  5. Controlling Genetic Drift through Negative Drift
  6. Runtime of the $r$-cGA on G-OneMax
  7. Improved Runtime Analysis of the $r$-cGA on $r$-OneMax
  8. Analysis in the Presence of Borders
  9. Analysis through Experimentation
  10. Conclusion
  11. Acknowledgements

Key Result

Theorem 1

Consider a stochastic process $(X_t, \mathcal{F}_t)_{t \in \mathbb{N}_0}$ with $X_0 \geq b$. Let $a\le b$. Let $T$ be a stopping time defined as Assume that, for all $t$, there are constants $0 < \varepsilon < (b-a)/2$ and $0 < c < b-a$ satisfying Then, for all $t \geq 0$,

Figures (2)

  • Figure 1: Empirical runtime of the $r$-cGA on $r$-OneMax and G-OneMax; with the top side showing results for $r$-OneMax at $r=3$ (left), $r=4$ (middle), and $r=5$ (right), and the bottom side for G-OneMax at $r=3$ (left), $r=4$ (middle), and $r=5$ (right); $n\in\{200,\dots,400\}$, $K\in\{100,\dots,1000\}$ and averaged over 100 runs.
  • Figure 2: The analysis of frequency evolution for the $r$-cGA on both $r$-OneMax and G-OneMax; with the left side displaying results for $r$-OneMax and the right side for G-OneMax; $n=400$ and $K=600$; $r\in\{0,\dots,7\}$, for a particular position.

Theorems & Definitions (13)

  • Theorem 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Theorem 6
  • Theorem 7
  • Lemma 8
  • Lemma 9
  • Lemma 10
  • ...and 3 more