Near Optimal Best Arm Identification for Clustered Bandits

This work investigates the problem of best arm identification for multi-agent multi-armed bandits. We consider $N$ agents grouped into $M$ clusters, where each cluster solves a stochastic bandit problem. The mapping between agents and bandits is a priori unknown. Each bandit is associated with $K$ arms, and the goal is to identify the best arm for each agent under a $\delta$-probably correct ($\delta$-PC) framework, while minimizing sample complexity and communication overhead. We propose two novel algorithms: Clustering then Best Arm Identification (Cl-BAI) and Best Arm Identification then Clustering (BAI-Cl). Cl-BAI uses a two-phase approach that first clusters agents based on the bandit problems they are learning, followed by identifying the best arm for each cluster. BAI-Cl reverses the sequence by identifying the best arms first and then clustering agents accordingly. Both algorithms leverage the successive elimination framework to ensure computational efficiency and high accuracy. We establish $\delta$-PC guarantees for both methods, derive bounds on their sample complexity, and provide a lower bound for this problem class. Moreover, when $M$ is small (a constant), we show that the sample complexity of a variant of BAI-Cl is minimax optimal in an order-wise sense. Experiments on synthetic and real-world datasets (MovieLens, Yelp) demonstrate the superior performance of the proposed algorithms in terms of sample and communication efficiency, particularly in settings where $M \ll N$.