Bandits with Multiple Optimal Arms: Minimax Regret and Non-Adaptivit

Abstract:We study multi-armed bandits (MAB) with multiple optimal arms, motivated by the fact that many practical decision making problems admit multiple correct answers. For $K$-armed bandits with $A$ optimal arms, we first provide a sharper analysis of previous sub-sampling algorithms (De Heide et al., 2021; Zhu and Nowak, 2020), establishing a $\tilde{O}\Big(\frac{K-A}{\sqrt{KA}}\sqrt{T} \Big)$ minimax regret, where $T$ is the total number of interactions and $\tilde O(\cdot)$ drops all constant and logarithmic factors, improving the previous $\tilde{O}(\sqrt{KT/A})$ regret. We then provide a matching lower bound up to logarithmic factors, indicating that our established rate is nearly minimax-optimal. We further show that the knowledge of $A$ up to $\tilde{O}(1)$ factors is necessary to achieve near-optimal regret, as near-optimal algorithms for one number of optimal arms must incur substantially larger regret than optimal regret for a smaller number. Overall, our results provide a comprehensive minimax characterization of $K$-armed bandits with $A$ over the entire range of $1 \leq A \leq K-1$.

Machine Learning (stat.ML); Artificial Intelligence (cs.AI); Machine Learning (cs.LG); Statistics Theory (math.ST); Methodology (stat.ME)

arXiv:2609.38659 [stat.ML]

(or arXiv:2609.38659v1 [stat.ML] for this version)

https://doi.org/10.48550/arXiv.2609.38659 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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