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Tight Runtime Guarantees From Understanding the Population Dynamics of the GSEMO Multi-Objective Evolutionary Algorithm

Benjamin Doerr, Martin Krejca, Andre Opris

TL;DR

This work advances the mathematical understanding of the GSEMO by focusing on its population dynamics on the COCZ benchmark. It introduces a modified GSEMO with dummy individuals to decouple population-size effects, enabling tight tail-b bounds and a clear two-phase runtime picture: a fast attainment of the Pareto front in $O(n^2)$ iterations followed by a slower expansion to cover the entire front in $\Omega(n^2 \log n)$ evaluations. The authors also extend the methodology to SEMO and to OMM and OJZJ benchmarks, obtaining matching lower bounds and, in the case of OJZJ$_k$, new bounds for $k\in\{2,3\}$. Overall, the paper offers a robust framework for analyzing MOEAs via population-dynamics and lays groundwork for analyzing more complex algorithms like NSGA-II.

Abstract

The global simple evolutionary multi-objective optimizer (GSEMO) is a simple, yet often effective multi-objective evolutionary algorithm (MOEA). By only maintaining non-dominated solutions, it has a variable population size that automatically adjusts to the needs of the optimization process. The downside of the dynamic population size is that the population dynamics of this algorithm are harder to understand, resulting, e.g., in the fact that only sporadic tight runtime analyses exist. In this work, we significantly enhance our understanding of the dynamics of the GSEMO, in particular, for the classic CountingOnesCountingZeros (COCZ) benchmark. From this, we prove a lower bound of order $Ω(n^2 \log n)$, for the first time matching the seminal upper bounds known for over twenty years. We also show that the GSEMO finds any constant fraction of the Pareto front in time $O(n^2)$, improving over the previous estimate of $O(n^2 \log n)$ for the time to find the first Pareto optimum. Our methods extend to other classic benchmarks and yield, e.g., the first $Ω(n^{k+1})$ lower bound for the OJZJ benchmark in the case that the gap parameter is $k \in \{2,3\}$. We are therefore optimistic that our new methods will be useful in future mathematical analyses of MOEAs.

Tight Runtime Guarantees From Understanding the Population Dynamics of the GSEMO Multi-Objective Evolutionary Algorithm

TL;DR

This work advances the mathematical understanding of the GSEMO by focusing on its population dynamics on the COCZ benchmark. It introduces a modified GSEMO with dummy individuals to decouple population-size effects, enabling tight tail-b bounds and a clear two-phase runtime picture: a fast attainment of the Pareto front in iterations followed by a slower expansion to cover the entire front in evaluations. The authors also extend the methodology to SEMO and to OMM and OJZJ benchmarks, obtaining matching lower bounds and, in the case of OJZJ, new bounds for . Overall, the paper offers a robust framework for analyzing MOEAs via population-dynamics and lays groundwork for analyzing more complex algorithms like NSGA-II.

Abstract

The global simple evolutionary multi-objective optimizer (GSEMO) is a simple, yet often effective multi-objective evolutionary algorithm (MOEA). By only maintaining non-dominated solutions, it has a variable population size that automatically adjusts to the needs of the optimization process. The downside of the dynamic population size is that the population dynamics of this algorithm are harder to understand, resulting, e.g., in the fact that only sporadic tight runtime analyses exist. In this work, we significantly enhance our understanding of the dynamics of the GSEMO, in particular, for the classic CountingOnesCountingZeros (COCZ) benchmark. From this, we prove a lower bound of order , for the first time matching the seminal upper bounds known for over twenty years. We also show that the GSEMO finds any constant fraction of the Pareto front in time , improving over the previous estimate of for the time to find the first Pareto optimum. Our methods extend to other classic benchmarks and yield, e.g., the first lower bound for the OJZJ benchmark in the case that the gap parameter is . We are therefore optimistic that our new methods will be useful in future mathematical analyses of MOEAs.
Paper Structure (21 sections, 27 theorems, 30 equations)

This paper contains 21 sections, 27 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Our contribution:
  3. Previous Works
  4. Preliminaries
  5. The Algorithms SEMO and GSEMO
  6. The COCZ Benchmark
  7. Mathematical Tools
  8. Runtime Analysis on COCZ
  9. Progress of the modified (G)SEMO on the $g_1$-values.
  10. Progress of the modified (G)SEMO on the $g_2$-values.
  11. How the original (G)SEMO computes the full Pareto front.
  12. Runtime Analysis on OMM and OJZJ
  13. Progress of the modified (G)SEMO.
  14. OMM
  15. OJZJ
  16. ...and 6 more sections

Key Result

Theorem 1

Let $k \in \mathbb{N}_{\geq 1}$, and let $\{D_i\}_{i \in [k]}$ be independent geometric random variables with respective positive success probabilities $(p_i)_{i \in [k]}$. Let $T^{\star} \coloneqq \sum_{i \in [k]} D_i$ , $s \coloneqq \sum_{i \in [k]} \frac{1}{p_i^2}$, and $p_{\min} \coloneqq \min \

Theorems & Definitions (39)

  • Theorem 1: Witt14
  • Theorem 2: CormenLRS01IntroductionToAlgorithms
  • Theorem 3: Chernoff52
  • Theorem 4
  • Lemma 4
  • Lemma 4
  • Lemma 4
  • Corollary 5
  • Lemma 5
  • Lemma 5
  • ...and 29 more