Analysing the Robustness of NSGA-II under Noise

TL;DR

This paper provides the first rigorous analysis of NSGA-II's robustness to noise in multiobjective optimization, showing it can outperform GSEMO under certain noisy conditions with a phase transition at noise probability 1/2.

Contribution

It demonstrates that NSGA-II can handle noise more effectively than GSEMO, with a detailed analysis of its performance and a phase transition point at noise probability 1/2.

Findings

NSGA-II handles noise efficiently when p<1/2.

GSEMO fails under noisy conditions, removing large parts of the population.

A phase transition at p=1/2 changes the runtime from polynomial to exponential.

Abstract

Runtime analysis has produced many results on the efficiency of simple evolutionary algorithms like the (1+1) EA, and its analogue called GSEMO in evolutionary multiobjective optimisation (EMO). Recently, the first runtime analyses of the famous and highly cited EMO algorithm NSGA-II have emerged, demonstrating that practical algorithms with thousands of applications can be rigorously analysed. However, these results only show that NSGA-II has the same performance guarantees as GSEMO and it is unclear how and when NSGA-II can outperform GSEMO. We study this question in noisy optimisation and consider a noise model that adds large amounts of posterior noise to all objectives with some constant probability $p$ per evaluation. We show that GSEMO fails badly on every noisy fitness function as it tends to remove large parts of the population indiscriminately. In contrast, NSGA-II is able to handle the noise efficiently on \textsc{LeadingOnesTrailingZeroes} when $p<1/2$, as the algorithm is able to preserve useful search points even in the presence of noise. We identify a phase transition at $p=1/2$ where the expected time to cover the Pareto front changes from polynomial to exponential. To our knowledge, this is the first proof that NSGA-II can outperform GSEMO and the first runtime analysis of NSGA-II in noisy optimisation.