Top-$k$ Pareto Bandits: Hypervolume Regret for Multi-Objective Slate Selection

Abstract:We consider a stochastic multi-objective bandit problem where, at each round, the agent selects a slate of $k$ arms and observes their $d$-dimensional reward vectors under semi-bandit feedback. We do not aim at identifying a single optimal arm; instead, we consider the problem of maintaining a small set of actions that jointly approximate the Pareto frontier. We formalize this objective through the dominated hypervolume induced by the selected subset of arms, and define an $\alpha$-approximate hypervolume regret with respect to the best size-$k$ subset achievable in hindsight, where $\alpha = 1 - 1/e$ reflects the approximation guarantee of greedy maximization for monotone submodular functions. To address this problem, we introduce \textit{THV-UCB}, an optimistic algorithm that selects arms greedily based on optimistic estimates of their marginal hypervolume contributions. We establish a gap-free regret bound $\tilde{O}(d\sqrt{nkT})$ that holds on every instance, together with a gap-dependent bound $\tilde{O}(nk^{2.5}/\Delta_{\min})$ that becomes polylogarithmic in $T$ once the arms are sufficiently well separated. Our results provide theoretical support for using small subsets to approximate Pareto fronts in various multi-objective applications.

Machine Learning (cs.LG); Artificial Intelligence (cs.AI); Machine Learning (stat.ML)

I.2.6; I.2.8; G.3; F.2.2

arXiv:2607.26273 [cs.LG]

(or arXiv:2607.26273v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2607.26273 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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