In this paper, we study Federated Bandit, a decentralized Multi-Armed Bandit problem with a set of N agents, who can only communicate their local data with neighbors described by a connected graph G. Each agent makes a sequence of decisions on selecting an arm from M candidates, yet they only have access to local and potentially biased feedback/evaluation of the true reward for each action taken. Learning only locally will lead agents to sub-optimal actions while converging to a no-regret

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... gy requires a collection of distributed data. Motivated by the proposal of federated learning, we aim for a solution with which agents will never share their local observations with a central entity, and will be allowed to only share a private copy of his/her own information with their neighbors. We first propose a decentralized bandit algorithm Gossip_UCB, which is a coupling of variants of both the classical gossiping algorithm and the celebrated Upper Confidence Bound (UCB) bandit algorithm. We show that Gossip_UCB successfully adapts local bandit learning into a global gossiping process for sharing information among connected agents, and achieves guaranteed regret at the order of O(max{poly(N,M) log T, poly(N,M)log_λ_2^-1 N}) for all N agents, where λ_2∈(0,1) is the second largest eigenvalue of the expected gossip matrix, which is a function of G. We then propose Fed_UCB, a differentially private version of Gossip_UCB, in which the agents preserve ϵ-differential privacy of their local data while achieving O(max{poly(N,M)/ϵlog^2.5 T, poly(N,M) (log_λ_2^-1 N + log T) }) regret.