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The First Theoretical Approximation Guarantees for the Non-Dominated Sorting Genetic Algorithm III (NSGA-III)

Renzhong Deng, Weijie Zheng, Benjamin Doerr

TL;DR

This paper delivers the first theoretical analysis of NSGA-III's ability to approximate the Pareto front on the bi-objective OneMinMax problem when the population size $N$ is smaller than the front size. It proves a constant-factor MEI bound $MEI\le \lceil\frac{(5-2\sqrt{2})\,n}{N_r-1}\rceil$ provided $N\ge N_r$, with an accompanying runtime bound $O(N\,n^c\log n)$ where $c=\lceil\frac{2(2-\sqrt{2})\,n}{N_r-1}\rceil$; the optimal MEI is $MEI_{opt}=\lceil\frac{n}{N-1}\rceil$, and the factor is roughly $2.17$. The analysis reveals a key insight: setting $N_r=N$ yields the best approximation, and increasing $N_r$ beyond $N$ can impair performance due to potential loss of extremal points. Experiments on OneMinMax and LOTZ confirm that $N_r=N$ achieves optimal MEI and often outperforms NSGA-II, while large $N_r$ can cause deterioration, providing practical guidance for NSGA-III configuration and highlighting directions for tighter theoretical bounds.

Abstract

This work conducts a first theoretical analysis studying how well the NSGA-III approximates the Pareto front when the population size $N$ is less than the Pareto front size. We show that when $N$ is at least the number $N_r$ of reference points, then the approximation quality, measured by the maximum empty interval (MEI) indicator, on the OneMinMax benchmark is such that there is no empty interval longer than $\lceil\frac{(5-2\sqrt2)n}{N_r-1}\rceil$. This bound is independent of $N$, which suggests that further increasing the population size does not increase the quality of approximation when $N_r$ is fixed. This is a notable difference to the NSGA-II with sequential survival selection, where increasing the population size improves the quality of the approximations. We also prove two results indicating approximation difficulties when $N<N_r$. These theoretical results suggest that the best setting to approximate the Pareto front is $N_r=N$. In our experiments, we observe that with this setting the NSGA-III computes optimal approximations, very different from the NSGA-II, for which optimal approximations have not been observed so far.

The First Theoretical Approximation Guarantees for the Non-Dominated Sorting Genetic Algorithm III (NSGA-III)

TL;DR

This paper delivers the first theoretical analysis of NSGA-III's ability to approximate the Pareto front on the bi-objective OneMinMax problem when the population size is smaller than the front size. It proves a constant-factor MEI bound provided , with an accompanying runtime bound where ; the optimal MEI is , and the factor is roughly . The analysis reveals a key insight: setting yields the best approximation, and increasing beyond can impair performance due to potential loss of extremal points. Experiments on OneMinMax and LOTZ confirm that achieves optimal MEI and often outperforms NSGA-II, while large can cause deterioration, providing practical guidance for NSGA-III configuration and highlighting directions for tighter theoretical bounds.

Abstract

This work conducts a first theoretical analysis studying how well the NSGA-III approximates the Pareto front when the population size is less than the Pareto front size. We show that when is at least the number of reference points, then the approximation quality, measured by the maximum empty interval (MEI) indicator, on the OneMinMax benchmark is such that there is no empty interval longer than . This bound is independent of , which suggests that further increasing the population size does not increase the quality of approximation when is fixed. This is a notable difference to the NSGA-II with sequential survival selection, where increasing the population size improves the quality of the approximations. We also prove two results indicating approximation difficulties when . These theoretical results suggest that the best setting to approximate the Pareto front is . In our experiments, we observe that with this setting the NSGA-III computes optimal approximations, very different from the NSGA-II, for which optimal approximations have not been observed so far.
Paper Structure (17 sections, 11 theorems, 3 equations, 3 figures, 2 tables, 3 algorithms)

This paper contains 17 sections, 11 theorems, 3 equations, 3 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. NSGA-III
  4. OneMinMax and Approximation Metric
  5. Approximation Guarantee when $N_r\le N$
  6. Reach Extremal Points
  7. Good Approximation Guarantee
  8. Possible Difficulty for Large $N_r>N$
  9. Lose Extremal Points
  10. Possible Bad Approximation Quality
  11. Experiments
  12. Experimental Settings
  13. Experimental Results
  14. The Impact of $N_r$
  15. The Actual Approximation Ability
  16. ...and 2 more sections

Key Result

Lemma 3

For all $N\in \mathbb{N}_{\ge 2}$, we have $\textsc{MEI}\xspace_{\rm{opt}\xspace}(N) = \lceil\frac{n}{N-1}\rceil$.

Figures (3)

  • Figure 1: Structured reference points set for a three-objective problem with p = 4 DebJ14.
  • Figure 2: The MEI for generations $[1..100]$ and $[3001..3100]$ after $T_{\rm{start}\xspace}$, in one exemplary run on the OneMinMax problem with $n=601$ and $N=76$.
  • Figure 3: The MEI for generations $[1..1000]$ after $T'_{\rm{start}\xspace}$, in one exemplary run on the LOTZ problem with $n=120$ and $N=16$.

Theorems & Definitions (13)

  • Definition 1: GielL10
  • Definition 2: ZhengD22geccoZhengD24approx
  • Lemma 3: ZhengD22geccoZhengD24approx
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • Lemma 9
  • Lemma 10
  • ...and 3 more