Analysing the robustness of nsga-ii under noise

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 LeadingOnes-TrailingZeroes when p < 1/2, as the algorithm is able to preserve useful search points. We identify a phase transition at p = 1/2 where the expected time to cover the Pareto front changes from polynomial to exponential. This is the first proof that NSGA-II can outperform GSEMO and the first runtime analysis of NSGA-II in noisy optimisation.