  We provide two distributed confidence ball algorithms for solving linear
bandit problems in peer to peer networks with limited communication
capabilities. For the first, we assume that all the peers are solving the same
linear bandit problem, and prove that our algorithm achieves the optimal
asymptotic regret rate of any centralised algorithm that can instantly
communicate information between the peers. For the second, we assume that there
are clusters of peers solving the same bandit problem within each cluster, and
we prove that our algorithm discovers these clusters, while achieving the
optimal asymptotic regret rate within each one. Through experiments on several
real-world datasets, we demonstrate the performance of proposed algorithms
compared to the state-of-the-art.
