A First Runtime Analysis of the PAES-25: An Enhanced Variant of the Pareto Archived Evolution Strategy

2025-07-04

AI Summary

This work addresses the runtime analysis of the multi-objective evolutionary algorithm PAES-25 on the LOTZ benchmark, a canonical bi-objective problem extended to arbitrary objectives. Method: We conduct the first rigorous theoretical analysis of PAES-25’s local search dynamics under one-bit and standard bit-wise mutation, combined with three archiving strategies: adaptive grid, hypervolume, and multi-layer grid. Contribution/Results: We derive tight bounds on the expected first-hitting time: Θ(n³) for m = 2 objectives; Θ(n³ log²n) for m = 4; and Θ(n(2n/m)^(m/2) log(n/m)) for m > 4, with an O(n⁴) bound for the bi-objective case. Notably, for m ≥ 4, our upper bound improves upon the previously known best upper bounds for (G)SEMO. The analysis systematically characterizes how distinct archiving mechanisms influence solution-set distribution quality, establishing a novel theoretical framework for understanding local search behavior in multi-objective evolutionary algorithms.

Technology Category

• Search and Optimization: Evolutionary Computation
• Machine Learning: Evolutionary Learning
• Multiagent Systems: Mechanism Design

Application Category

• Web Mining and Content Analysis: Models for Web evolution
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

This paper presents a first mathematical runtime analysis of PAES-25, an enhanced version of the original Pareto Archived Evolution Strategy (PAES) coming from the study of telecommunication problems over two decades ago to understand the dynamics of local search of MOEAs on many-objective fitness landscapes. We derive tight expected runtime bounds of PAES-25 with one-bit mutation on $m$-LOTZ until the entire Pareto front is found: $Θ(n^3)$ iterations if $m=2$, $Θ(n^3 log^2(n))$ iterations if $m=4$ and $Θ(n(2n/m)^{m/2} log(n/m))$ iterations if $m>4$ where $n$ is the problem size and $m$ the number of objectives. To the best of our knowledge, these are the first known tight runtime bounds for an MOEA outperforming the best known upper bound of $O(n^{m+1})$ for (G)SEMO on $m$-LOTZ when $m$ is at least $4$. We also show that archivers, such as the Adaptive Grid Archiver (AGA), Hypervolume Archiver (HVA) or Multi-Level Grid Archiver (MGA), help to distribute the set of solutions across the Pareto front of $m$-LOTZ efficiently. We also show that PAES-25 with standard bit mutation optimizes the bi-objective LOTZ benchmark in expected $O(n^4)$ iterations, and we discuss its limitations on other benchmarks such as OMM or COCZ.

Problem

• Analyze runtime of PAES-25 on many-objective landscapes
• Derive tight bounds for PAES-25 on m-LOTZ problem
• Evaluate archivers' role in Pareto front distribution

Innovation

• Enhanced PAES-25 with one-bit mutation
• Tight runtime bounds for m-LOTZ
• Efficient archivers distribute solutions