Local Optima in Diversity Optimization: Non-trivial Offspring Population is Essential

TL;DR

This work investigates diversity optimization (EDO) and reveals that elitist $(\mu+1)$ schemes can get trapped in locally optimal populations, unable to improve diversity by single-individual replacements. It introduces two population-space approaches, the $(\mu+\lambda)$ EA_D with $\lambda\ge\mu$ and the $(1_\mu+1_\mu)$ EA_D, augmented with a jump-and-repair mutation to achieve optimally diverse $k$-vertex covers. A key theoretical contribution is a concrete instance demonstrating local optima in the population space, along with a runtime analysis showing that large-offspring schemes can reach optimal diversity with probability at least $2^{-k\mu}(1-o(1))$ per iteration, yielding $O(2^{k+\mu})$ iterations when $k\mu=o(\sqrt{n})$ and potentially polynomial-time when $k=O(\log n)$. The results inform how to design diversity-focused EAs and highlight the necessity of multi-offspring updates to escape population-space local optima, with implications for broader quality-diversity optimization tasks.

Abstract

The main goal of diversity optimization is to find a diverse set of solutions which satisfy some lower bound on their fitness. Evolutionary algorithms (EAs) are often used for such tasks, since they are naturally designed to optimize populations of solutions. This approach to diversity optimization, called EDO, has been previously studied from theoretical perspective, but most studies considered only EAs with a trivial offspring population such as the $(μ+ 1)$ EA. In this paper we give an example instance of a $k$-vertex cover problem, which highlights a critical difference of the diversity optimization from the regular single-objective optimization, namely that there might be a locally optimal population from which we can escape only by replacing at least two individuals at once, which the $(μ+ 1)$ algorithms cannot do. We also show that the $(μ+ λ)$ EA with $λ\ge μ$ can effectively find a diverse population on $k$-vertex cover, if using a mutation operator inspired by Branson and Sutton (TCS 2023). To avoid the problem of subset selection which arises in the $(μ+ λ)$ EA when it optimizes diversity, we also propose the $(1_μ+ 1_μ)$ EA$_D$, which is an analogue of the $(1 + 1)$ EA for populations, and which is also efficient at optimizing diversity on the $k$-vertex cover problem.

Figures (1)

• Figure 1: Graph $G$, for which we have a population of 4-covers with suboptimal diversity, from which it is impossible to escape by replacing only one individual.

Theorems & Definitions (10)

• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Theorem 1
• proof