A First Runtime Analysis of NSGA-III on a Many-Objective Multimodal Problem: Provable Exponential Speedup via Stochastic Population Update

• URL: http://arxiv.org/abs/2505.01256v3
• Date: Wed, 21 May 2025 16:20:56 GMT
• Title: A First Runtime Analysis of NSGA-III on a Many-Objective Multimodal Problem: Provable Exponential Speedup via Stochastic Population Update
• Authors: Andre Opris,
• Abstract summary: NSGA-III is a prominent algorithm in evolutionary many-objective optimization.<n>This paper conducts a rigorous runtime analysis of NSGA-III on the many-objective $OJZJfull$ benchmark.<n>We show that NSGA-III is faster than NSGA-II by a factor of $mu/nd/2$ for some $mu in omega(nd/2ln)$.
• Score: 1.223779595809275
• License: http://creativecommons.org/publicdomain/zero/1.0/
• Abstract: The NSGA-III is a prominent algorithm in evolutionary many-objective optimization. It is well-suited for optimizing functions with more than three objectives, setting it apart from the classic NSGA-II. However, theoretical insights about NSGA-III of when and why it performs well are still in its early development. This paper addresses this point and conducts a rigorous runtime analysis of NSGA-III on the many-objective $\OJZJfull$ benchmark ($\OJZJ$ for short), providing runtime bounds where the number of objectives is constant. We show that NSGA-III finds the Pareto front of $\OJZJ$ in time $O(n^{k+d/2}+ \mu n \ln(n))$ where $n$ is the problem size, $d$ is the number of objectives, $k$ is the gap size, a problem specific parameter, if its population size $\mu \in 2^{O(n)}$ is at least $(2n/d+1)^{d/2}$. Notably, NSGA-III is faster than NSGA-II by a factor of $\mu/n^{d/2}$ for some $\mu \in \omega(n^{d/2})$. We also show that a stochastic population update, proposed by~\citet{UpBian}, provably guarantees a speedup of order $\Theta((k/b)^{k-1})$ in the runtime where $b>0$ is a constant. Besides~\cite{DoerrNearTight}, this is the first rigorous runtime analysis of NSGA-III on \OJZJ. Proving these bounds requires a much deeper understanding of the population dynamics of NSGA-III than previous papers achieved.

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