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.

翻译：运行时分析已为简单进化算法（如 (1+1) EA 及其在进化多目标优化（EMO）中的对应算法 GSEMO）的效率提供了大量结果。近期，对著名且被广泛引用的 EMO 算法 NSGA-II 的首次运行时分析已经出现，表明这些具有数千次应用的实用算法可以得到严格分析。然而，这些结果仅表明 NSGA-II 与 GSEMO 具有相同的性能保证，但尚不清楚 NSGA-II 如何以及何时能够优于 GSEMO。我们在含噪声优化中研究此问题，并考虑一种噪声模型，该模型以每次评估的恒定概率 $p$ 向所有目标添加大量后验噪声。我们证明，GSEMO 在任意含噪声适应度函数上都会严重失败，因为它倾向于不加区分地移除大部分种群。相比之下，当 $p<1/2$ 时，NSGA-II 能够高效处理 \textsc{LeadingOnesTrailingZeroes} 问题上的噪声，因为该算法即使在存在噪声的情况下也能保留有用的搜索点。我们发现在 $p=1/2$ 处存在一个相变，此时覆盖帕累托前沿的期望时间从多项式变为指数级。据我们所知，这是首次证明 NSGA-II 可以优于 GSEMO，也是首次对 NSGA-II 在含噪声优化中的运行时分析。