Constrained Pareto Set Identification with Bandit Feedback

TL;DR

This paper introduces a new algorithm for efficiently identifying the Pareto Set under constraints in a multivariate bandit setting, with theoretical guarantees and empirical validation showing near-optimal performance.

Contribution

The paper proposes a novel fixed-confidence algorithm for constrained Pareto Set identification that outperforms existing methods and matches theoretical lower bounds.

Findings

The algorithm significantly reduces sample complexity compared to racing-like algorithms.

Theoretical lower bounds demonstrate near-optimality of the proposed approach.

Empirical results confirm the effectiveness across benchmark problems.

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.