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This paper introduces SUAK, an adaptive algorithm for bandits with anytime knapsacks that optimizes reward accumulation under continuous cost constraints by strategically managing budget utilization. The authors demonstrate theoretically and via simulation that this approach achieves optimal regret bounds while outperforming naive adaptations of standard multi-armed bandit algorithms.
Bandits with Anytime Knapsacks
arXiv:2501.18560v2 Announce Type: replace Abstract: We consider bandits with anytime knapsacks (BwAK), a novel version of the BwK problem where there is an \textit{anytime} cost constraint instead of a total cost budget. This problem setting introduces additional complexities as it mandates adherence to the constraint throughout the decision-making process. We propose SUAK (Strategic Under-utilization for Anytime Knapsacks), an algorithm that utilizes upper confidence bounds to identify the optimal mixture of arms while maintaining a balance between exploration and exploitation. SUAK is an adaptive algorithm that strategically utilizes the available budget in each round in the decision-making process and skips a round when it is possible to violate the anytime cost constraint. In particular, SUAK slightly under-utilizes the available cost budget to reduce the need for skipping rounds. We show that SUAK attains the same problem-dependent regret upper bound of $ O(K \log T)$ established in prior work under the simpler BwK framework, where $K$ denotes the number of arms, and $T$ denotes the time horizon. We also establish an instance-dependent lower bound showing that the $O(K\log T)$ dependence is unavoidable. Finally, we show via extensive simulations that SUAK achieves considerably higher rewards than naive adaptations of BwK algorithms that do not optimize learning under anytime cost constraints.
Abstract:We consider bandits with anytime knapsacks (BwAK), a novel version of the BwK problem where there is an \textit{anytime} cost constraint instead of a total cost budget. This problem setting introduces additional complexities as it mandates adherence to the constraint throughout the decision-making process. We propose SUAK (Strategic Under-utilization for Anytime Knapsacks), an algorithm that utilizes upper confidence bounds to identify the optimal mixture of arms while maintaining a balance between exploration and exploitation. SUAK is an adaptive algorithm that strategically utilizes the available budget in each round in the decision-making process and skips a round when it is possible to violate the anytime cost constraint. In particular, SUAK slightly under-utilizes the available cost budget to reduce the need for skipping rounds. We show that SUAK attains the same problem-dependent regret upper bound of $ O(K \log T)$ established in prior work under the simpler BwK framework, where $K$ denotes the number of arms, and $T$ denotes the time horizon. We also establish an instance-dependent lower bound showing that the $O(K\log T)$ dependence is unavoidable. Finally, we show via extensive simulations that SUAK achieves considerably higher rewards than naive adaptations of BwK algorithms that do not optimize learning under anytime cost constraints.
Machine Learning (cs.LG)
arXiv:2501.18560 [cs.LG]
(or arXiv:2501.18560v2 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2501.18560 Focus to learn more arXiv-issued DOI via DataCite
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