Adaptive Resampling with Bootstrap for Noisy Multi-Objective Optimization Problems

TL;DR

This work tackles noisy multi-objective optimization by balancing exploration and resampling through a bootstrap-based adaptive resampling strategy guided by the probability of domination. It integrates bootstrap estimates of mean distributions with a domination-based decision rule into NSGA-II, enabling selective resampling of promising points while conserving budget. Across simulations with Gaussian and chi-square noise, the proposed ARB method generally outperforms several baselines, though performance varies with the noise distribution. The approach is notable for its adaptivity, distribution-free domination estimation, and potential applicability to univariate problems beyond MOOPs.

Abstract

The challenge of noisy multi-objective optimization lies in the constant trade-off between exploring new decision points and improving the precision of known points through resampling. This decision should take into account both the variability of the objective functions and the current estimate of a point in relation to the Pareto front. Since the amount and distribution of noise are generally unknown, it is desirable for a decision function to be highly adaptive to the properties of the optimization problem. This paper presents a resampling decision function that incorporates the stochastic nature of the optimization problem by using bootstrapping and the probability of dominance. The distribution-free estimation of the probability of dominance is achieved using bootstrap estimates of the means. To make the procedure applicable even with very few observations, we transfer the distribution observed at other decision points. The efficiency of this resampling approach is demonstrated by applying it in the NSGA-II algorithm with a sequential resampling procedure under multiple noise variations.

Figures (5)

• Figure 1: Fraction of performing best in terms of dominated hypervolume of the different clusters of resampling strategies by using optimized parameters for each setting. The boxplots are representing the distribution of the fraction based on 100 random splits between selection replications and comparison replications. For the sake
• Figure 2: Average dominated Hypervolume over 30 replication with 50.000 evaluations based on a parameter optimization step with 5000 evaluations in the setting with Gaussian noise.
• Figure 3: Average dominated Hypervolume over 30 replication with 50.000 evaluations based on a parameter optimization step with 5000 evaluations in the setting with $\chi^2$ noise with df = 1.
• Figure 4: Average dominated Hypervolume over 30 replication with 50.000 evaluations based on a parameter optimization step with 5000 evaluations in the setting with $\chi^2$ noise with df = 2.
• Figure 5: Fraction of performing best of the different clusters of resampling strategies by using just one parameter setting for all problems.