Scalable Speed-ups for the SMS-EMOA from a Simple Aging Strategy
Mingfeng Li, Weijie Zheng, Benjamin Doerr
TL;DR
MOEAs are typically greedy in survival, motivating the authors to introduce an aging-based non-elitist survival mechanism for SMS-EMOA. The aging strategy gives new individuals a grace period of $\tau$ generations, enabling broader exploration and preserving Pareto-front points, which enables rigorous runtime analysis. The authors prove scalable speed-ups: for bi-objective OJZJ, $O\big(n^{k+1}/\Theta(k)^{k-1}\big)$ iterations, and for mOJZJ, $O\big(\overline{M}kmn^k/\Theta(k)^k\big)$ iterations, achieving $\max\{1,\Theta(k)^{k-1}\}$ speed-up over the classic SMS-EMOA across all objective counts. Empirical results on OJZJ and mOJZJ corroborate substantial speed-ups compared to both the original and stochastic-update variants, suggesting aging-based non-elitist strategies can yield scalable gains in MOEAs.
Abstract
Different from single-objective evolutionary algorithms, where non-elitism is an established concept, multi-objective evolutionary algorithms almost always select the next population in a greedy fashion. In the only notable exception, Bian, Zhou, Li, and Qian (IJCAI 2023) proposed a stochastic selection mechanism for the SMS-EMOA and proved that it can speed up computing the Pareto front of the bi-objective jump benchmark with problem size $n$ and gap parameter $k$ by a factor of $\max\{1,2^{k/4}/n\}$. While this constitutes the first proven speed-up from non-elitist selection, suggesting a very interesting research direction, it has to be noted that a true speed-up only occurs for $k \ge 4\log_2(n)$, where the runtime is super-polynomial, and that the advantage reduces for larger numbers of objectives as shown in a later work. In this work, we propose a different non-elitist selection mechanism based on aging, which exempts individuals younger than a certain age from a possible removal. This remedies the two shortcomings of stochastic selection: We prove a speed-up by a factor of $\max\{1,Θ(k)^{k-1}\}$, regardless of the number of objectives. In particular, a positive speed-up can already be observed for constant $k$, the only setting for which polynomial runtimes can be witnessed. Overall, this result supports the use of non-elitist selection schemes, but suggests that aging-based mechanisms can be considerably more powerful than stochastic selection mechanisms.