Multiple-play Stochastic Bandits with Prioritized Arm Capacity Sharing

This paper proposes a variant of multiple-play stochastic bandits tailored to resource allocation problems arising from LLM applications, edge intelligence, etc. The model is composed of $M$ arms and $K$ plays. Each arm has a stochastic number of capacities, and each unit of capacity is associated with a reward function. Each play is associated with a priority weight. When multiple plays compete for the arm capacity, the arm capacity is allocated in a larger priority weight first manner. Instance independent and instance dependent regret lower bounds of $\Omega( \alpha_1 \sigma \sqrt{KM T} )$ and $\Omega(\alpha_1 \sigma^2 \frac{M}{\Delta} \ln T)$ are proved, where $\alpha_1$ is the largest priority weight and $\sigma$ characterizes the reward tail. When model parameters are given, we design an algorithm named \texttt{MSB-PRS-OffOpt} to locate the optimal play allocation policy with a computational complexity of $O(MK^3)$. Utilizing \texttt{MSB-PRS-OffOpt} as a subroutine, an approximate upper confidence bound (UCB) based algorithm is designed, which has instance independent and instance dependent regret upper bounds matching the corresponding lower bound up to factors of $\sqrt{K \ln KT }$ and $\alpha_1 K^2$ respectively. To this end, we address nontrivial technical challenges arising from optimizing and learning under a special nonlinear combinatorial utility function induced by the prioritized resource sharing mechanism.