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

• URL: http://arxiv.org/abs/2504.21552v1
• Date: Wed, 30 Apr 2025 11:49:42 GMT
• Title: The First Theoretical Approximation Guarantees for the Non-Dominated Sorting Genetic Algorithm III (NSGA-III)
• Authors: Renzhong Deng, Weijie Zheng, Benjamin Doerr,
• Abstract summary: 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 $lceilfrac(5-2sqrt2)nN_r-1rceil$.<n>This is a notable difference to the NSGA-II with sequential survival selection, where increasing the population size improves the quality of the approximations.
• Score: 12.731778729620478
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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