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 O~(dnkT) that holds on every instance, together with a gap-dependent bound O~(nk2.5/Δmin) 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.
Identifying Pareto optimal solutions is critical to support multi-objective decision-making. We introduce the first anytime Multi-Objective Multi-Armed Bandit algorithm for the Pareto Set Identification problem, taking a Bayesian approach: Top-Two Pareto Front Thompson Sampling (TTPFTS). We benchmark TTPFTS against state-of-the-art fixed-budget Pareto Set Identification algorithms on synthetic environments. Next, we demonstrate its practical utility in a challenging multi-objective molecular discovery setting by efficiently exploring an ultra-large synthesis-on-demand molecular library. Furthermore, we introduce a novel uncertainty quantification metric that estimates our algorithm's confidence in the predicted Pareto set. We demonstrate that this metric effectively proxies true performance, yielding a robust methodology for monitoring learning progress in complex settings. Finally, we complement these empirical findings with a theoretical proof of the algorithm's asymptotic correctness.
Selective ensemble for modern machine learning systems requires choosing promising model candidates under limited evaluation budgets, while downstream tasks often specify only partial preferences over capabilities such as accuracy, robustness, and reasoning. This setting naturally gives rise to a sequential decision problem under partially specified linear preferences. We formalize it as preference-directed multi-objective bandits (PDMOB), where admissible trade-offs are represented by a polyhedral preference cone. Based on this formulation, we introduce Pareto C-optimality, which recovers standard Pareto optimality and single-weight scalarization as special cases. We then propose the preference-directed upper confidence bound (PrefUCB) algorithm, which maintains directional confidence intervals to guide exploration. We analyze both indicator-based and gap-weighted regret, and establish instance-dependent logarithmic bounds for both criteria, recovering the optimal logarithmic dependence on the horizon T in classical special cases. Experiments on large pre-trained model selective ensemble tasks and online asset allocation under institutional mandates validate the efficacy of our method.
Scalarization is widely used in multi-objective optimization owing to its simplicity and scalability. In many applications, the goal is to generate solutions that represent diverse user preferences, ideally with uniform coverage of the Pareto front (PF). However, uniformly sampling scalarization weights usually induces non-uniform coverage of the PF. We explain this mismatch through a geometric analysis of the scalarization path. As the scalarization weight varies, the corresponding solutions trace the PF with a generally non-uniform traversal speed. This speed induces an arc-length cumulative distribution function (CDF); inverting this CDF map yields a principled rule for selecting weights that produce uniform PF coverage. Building on this insight, we propose SURF (Sampling Uniformly along the PaReto Front). For structured problems, including bi-objective bandits, we derive closed-form expressions for this CDF map and the resulting PF-aware weight sampling rule. For general problems, SURF alternates between CDF reconstruction and weight sampling. Theoretically, we show that under provable conditions, SURF converges linearly to an unavoidable finite-sampling floor. Empirically, experiments on bandits, multi-objective-gymnasium, and multi-objective LLM alignment demonstrate that SURF efficiently achieves more uniform PF coverage than baselines.