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Approximation Guarantees for the Non-Dominated Sorting Genetic Algorithm II (NSGA-II)

Weijie Zheng, Benjamin Doerr

TL;DR

This paper analyzes how NSGA-II approximates the Pareto front when the population size is smaller than the front size, revealing that the standard selection based on the initial crowding distance can create sizable gaps. It introduces two robust alternatives—the NSGA-II with per-removal crowding-distance updates (current crowding distance) and the steady-state NSGA-II—and proves strong approximation guarantees for both on the OneMinMax benchmark, demonstrating that the largest empty interval on the front remains within a constant factor of the theoretical minimum. The authors connect MEI to ε-dominance and hypervolume to situate their measure among established metrics, and corroborate the theory with experiments showing superior and more stable approximation performance for the two proposed variants. The findings suggest practical guidance for MOEAs when resources constrain population size and point to promising avenues for extending these guarantees to other benchmarks and selection schemes.

Abstract

Recent theoretical works have shown that the NSGA-II efficiently computes the full Pareto front when the population size is large enough. In this work, we study how well it approximates the Pareto front when the population size is smaller. For the OneMinMax benchmark, we point out situations in which the parents and offspring cover well the Pareto front, but the next population has large gaps on the Pareto front. Our mathematical proofs suggest as reason for this undesirable behavior that the NSGA-II in the selection stage computes the crowding distance once and then removes individuals with smallest crowding distance without considering that a removal increases the crowding distance of some individuals. We then analyze two variants not prone to this problem. For the NSGA-II that updates the crowding distance after each removal (Kukkonen and Deb (2006)) and the steady-state NSGA-II (Nebro and Durillo (2009)), we prove that the gaps in the Pareto front are never more than a small constant factor larger than the theoretical minimum. This is the first mathematical work on the approximation ability of the NSGA-II and the first runtime analysis for the steady-state NSGA-II. Experiments also show the superior approximation ability of the two NSGA-II variants.

Approximation Guarantees for the Non-Dominated Sorting Genetic Algorithm II (NSGA-II)

TL;DR

This paper analyzes how NSGA-II approximates the Pareto front when the population size is smaller than the front size, revealing that the standard selection based on the initial crowding distance can create sizable gaps. It introduces two robust alternatives—the NSGA-II with per-removal crowding-distance updates (current crowding distance) and the steady-state NSGA-II—and proves strong approximation guarantees for both on the OneMinMax benchmark, demonstrating that the largest empty interval on the front remains within a constant factor of the theoretical minimum. The authors connect MEI to ε-dominance and hypervolume to situate their measure among established metrics, and corroborate the theory with experiments showing superior and more stable approximation performance for the two proposed variants. The findings suggest practical guidance for MOEAs when resources constrain population size and point to promising avenues for extending these guarantees to other benchmarks and selection schemes.

Abstract

Recent theoretical works have shown that the NSGA-II efficiently computes the full Pareto front when the population size is large enough. In this work, we study how well it approximates the Pareto front when the population size is smaller. For the OneMinMax benchmark, we point out situations in which the parents and offspring cover well the Pareto front, but the next population has large gaps on the Pareto front. Our mathematical proofs suggest as reason for this undesirable behavior that the NSGA-II in the selection stage computes the crowding distance once and then removes individuals with smallest crowding distance without considering that a removal increases the crowding distance of some individuals. We then analyze two variants not prone to this problem. For the NSGA-II that updates the crowding distance after each removal (Kukkonen and Deb (2006)) and the steady-state NSGA-II (Nebro and Durillo (2009)), we prove that the gaps in the Pareto front are never more than a small constant factor larger than the theoretical minimum. This is the first mathematical work on the approximation ability of the NSGA-II and the first runtime analysis for the steady-state NSGA-II. Experiments also show the superior approximation ability of the two NSGA-II variants.
Paper Structure (23 sections, 17 theorems, 35 equations, 1 figure, 1 table, 3 algorithms)

This paper contains 23 sections, 17 theorems, 35 equations, 1 figure, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Bi-objective Optimization and the OneMinMax Benchmark
  4. The Non-Dominated Sorting Genetic Algorithm II (NSGA-II)
  5. Approximation Measures
  6. Maximal Empty Interval Size
  7. $\varepsilon$-Dominance
  8. Background and Definition
  9. Relation of Maximal Empty Intervals and $\varepsilon$-Approximations
  10. Hypervolume
  11. Background and Definition
  12. Relation of Maximal Empty Intervals and Hypervolumes
  13. Difficulties for the NSGA-II to Approximate the Pareto Front
  14. Working With the Current Crowding Distance
  15. Implementation of the NSGA-II Using the Current Crowding Distance
  16. ...and 8 more sections

Key Result

Lemma 2

For all $N\in \mathbb{N}_{\ge 2}$, we have $\mathop{\mathrm{MEI}}\nolimits_{\mathop{\mathrm{opt}}\nolimits}(N)=\lceil \frac{n}{N-1}\rceil$.

Figures (1)

  • Figure 1: The $\mathop{\mathrm{MEI}}\nolimits$ for generations $[1..100]$ and $[3001..3100]$ after the two extreme points were found, in one exemplary run. For the steady-state NSGA-II with population size $N$, at point $i$ on the x-axis actually the data for generations $[1+N(i-1)..Ni$ are displayed, namely the maximal, minimal, and median MEI value in this interval (which always happened to be identical, and also identical to the current-CD value for $N=301$).

Theorems & Definitions (30)

  • Definition 1
  • Lemma 2
  • proof
  • Definition 3: (Multiplicative) $\varepsilon$-dominance LaumannsTDZ02
  • Lemma 4
  • proof
  • Corollary 5
  • Lemma 6
  • proof
  • Corollary 7
  • ...and 20 more