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

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 \textsc{OneJumpZeroJump} benchmark (\textsc{OjZj} for short), providing the first runtime bounds where the number of objectives is constant. We show that NSGA-III finds the Pareto front of \textsc{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 Bian et al., provably guarantees a speedup of order $\Theta((k/b)^{k-1})$ in the runtime where $b>0$ is a constant. To our knowledge, this is the first rigorous runtime analysis of NSGA-III on \textsc{OjZj}. Proving these bounds requires a much deeper understanding of the population dynamics of NSGA-III than previous papers achieved.