Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Sumit Adak, Carsten Witt

TL;DR

This work analyzes the runtime of the univariate, multi-valued EDAs framework by focusing on the $r$-valued compact genetic algorithm ($r$-cGA) performing optimization on the $r$-OneMax problem. By modeling the sampling frequencies and dissecting update dynamics into random-walk and biased steps, the authors establish a drift- and concentration-based analysis that bounds genetic drift and characterizes progress toward the optimum. Their main results show that, with high probability, the $r$-cGA solves the $r$-OneMax problem in $O\left(K \sqrt{n} \log r \log n\right)$ time for suitable $K$, and in $O\left(r^2 n \log^2 r \log^3 n\right)$ time under a different scaling of $K$, highlighting how parameter choices influence performance in multi-valued domains. Experiments on $n=500$ and varying $r$ corroborate the qualitative phase of the performance and motivate conjectures for the G-OneMax variant, indicating practical relevance and directions for further refinement of drift-aware analyses.

Abstract

A class of metaheuristic techniques called estimation-of-distribution algorithms (EDAs) are employed in optimization as more sophisticated substitutes for traditional strategies like evolutionary algorithms. EDAs generally drive the search for the optimum by creating explicit probabilistic models of potential candidate solutions through repeated sampling and selection from the underlying search space. Most theoretical research on EDAs has focused on pseudo-Boolean optimization. Jedidia et al. (GECCO 2023) proposed the first EDAs for optimizing problems involving multi-valued decision variables. By building a framework, they have analyzed the runtime of a multi-valued UMDA on the r-valued LeadingOnes function. Using their framework, here we focus on the multi-valued compact genetic algorithm (r-cGA) and provide a first runtime analysis of a generalized OneMax function. To prove our results, we investigate the effect of genetic drift and progress of the probabilistic model towards the optimum. After finding the right algorithm parameters, we prove that the r-cGA solves this r-valued OneMax problem efficiently. We show that with high probability, the runtime bound is O(r2 n log2 r log3 n). At the end of experiments, we state one conjecture related to the expected runtime of another variant of multi-valued OneMax function.

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

TL;DR

This work analyzes the runtime of the univariate, multi-valued EDAs framework by focusing on the -valued compact genetic algorithm (-cGA) performing optimization on the -OneMax problem. By modeling the sampling frequencies and dissecting update dynamics into random-walk and biased steps, the authors establish a drift- and concentration-based analysis that bounds genetic drift and characterizes progress toward the optimum. Their main results show that, with high probability, the -cGA solves the -OneMax problem in time for suitable , and in time under a different scaling of , highlighting how parameter choices influence performance in multi-valued domains. Experiments on and varying corroborate the qualitative phase of the performance and motivate conjectures for the G-OneMax variant, indicating practical relevance and directions for further refinement of drift-aware analyses.

Abstract

A class of metaheuristic techniques called estimation-of-distribution algorithms (EDAs) are employed in optimization as more sophisticated substitutes for traditional strategies like evolutionary algorithms. EDAs generally drive the search for the optimum by creating explicit probabilistic models of potential candidate solutions through repeated sampling and selection from the underlying search space. Most theoretical research on EDAs has focused on pseudo-Boolean optimization. Jedidia et al. (GECCO 2023) proposed the first EDAs for optimizing problems involving multi-valued decision variables. By building a framework, they have analyzed the runtime of a multi-valued UMDA on the r-valued LeadingOnes function. Using their framework, here we focus on the multi-valued compact genetic algorithm (r-cGA) and provide a first runtime analysis of a generalized OneMax function. To prove our results, we investigate the effect of genetic drift and progress of the probabilistic model towards the optimum. After finding the right algorithm parameters, we prove that the r-cGA solves this r-valued OneMax problem efficiently. We show that with high probability, the runtime bound is O(r2 n log2 r log3 n). At the end of experiments, we state one conjecture related to the expected runtime of another variant of multi-valued OneMax function.
Paper Structure (7 sections, 9 theorems, 35 equations, 1 algorithm)

This paper contains 7 sections, 9 theorems, 35 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries
  4. The Framework
  5. Genetic Drift for the $r$-valued cGA
  6. Runtime Analysis
  7. Experiments

Key Result

lemma thmcounterlemma

Let $f$ be an $r$-valued position and $i\in\{1,\dots,n\}$ be a neutral position of $f$. Consider the $r$-cGA without margins optimizing $f$. Then, for each $j\in\{0,\dots, r-1\}$, the frequencies $(p^{(t)}_{i,j})_{t\in\mathbb{N}}$ are a martingale.

Theorems & Definitions (16)

  • lemma thmcounterlemma
  • proof
  • theorem thmcountertheorem
  • theorem thmcountertheorem
  • proof
  • theorem thmcountertheorem
  • proof
  • theorem thmcountertheorem
  • proof
  • theorem thmcountertheorem
  • ...and 6 more