Constrained Pareto Set Identification with Bandit Feedback

Abstract: In this paper, we address the problem of identifying the Pareto Set under feasibility constraints in a multivariate bandit setting. Specifically, given a $K$-armed bandit with unknown means $\mu_1, \dots, \mu_K \in \mathbb{R}^d$, the goal is to identify the set of arms whose mean is not uniformly worse than that of another arm (i.e., not smaller for all objectives), while satisfying some known set of linear constraints, expressing, for example, some minimal performance on each objective. Our focus lies in fixed-confidence identification, for which we introduce an algorithm that significantly outperforms racing-like algorithms and the intuitive two-stage approach that first identifies feasible arms and then their Pareto Set. We further prove an information-theoretic lower bound on the sample complexity of any algorithm for constrained Pareto Set identification, showing that the sample complexity of our approach is near-optimal. Our theoretical results are supported by an extensive empirical evaluation on a series of benchmarks.