Likelihood-Free Multi-Objective Bayesian Optimization under Heteroscedastic Noise

Multi-objective Bayesian optimization (MOBO) fits a probabilistic surrogate, typically one Gaussian process (GP) per objective, so every acquisition function inherits the noise model of that surrogate. When the noise is heteroscedastic, that model cannot be identified from a single evaluation per design, and both the search and its evaluation degrade. We propose likelihood-free acquisition functions that do not require a noise model. We define a soft non-dominance indicator by nonparametric subsampling of the noisy observed objective vectors, derive it in closed form, and use it to weight a classification objective, which yields the likelihood-free counterparts of two state-of-the-art noisy acquisitions. These acquisition functions depend on the data only through the dominance relations between observations, and we prove that they preserve dominance and that their maximizers eventually lie in the true Pareto set. We further show that the standard evaluation protocols do not penalize a method for the set it would recommend, and we replace it by an exact decomposition of the decision maker's loss into sampling, set-selection and mis-selection terms, of which we observe mis-selection accounts for more than 40% at high noise. Across a comprehensive experimental campaign, GP-based baselines remain competitive when the noise is low and well specified, but as the noise level and structural complexity grow our acquisitions outperform GP baselines by margins that widen with the input dimension.