Analysing the Robustness of NSGA-II under Noise

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.