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arXiv · 2607.26273

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

Nicolas Gutowski·Fabien Chhel·Alexandre Letard·Sylvain Lamprier
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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 $α$-approximate hypervolume regret with respect to the best size-$k$ subset achievable in hindsight, where $α= 1 - 1/e$ 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 $\tilde{O}(nk^{2.5}/Δ_{\min})$ 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.

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Nicolas Gutowski, Fabien Chhel, Alexandre Letard, Sylvain Lamprier. 2026-07-28. Top-$k$ Pareto Bandits: Hypervolume Regret for Multi-Objective Slate Selection. https://arxiv.org/abs/2607.26273

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