  In this paper, we study \emph{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 strategy 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 $O(\max\{
\texttt{poly}(N,M) \log T, \texttt{poly}(N,M)\log_{\lambda_2^{-1}} N\})$ for
all $N$ agents, where $\lambda_2\in(0,1)$ 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 $\epsilon$-differential privacy of their local data while achieving
$O(\max \{\frac{\texttt{poly}(N,M)}{\epsilon}\log^{2.5} T, \texttt{poly}(N,M)
(\log_{\lambda_2^{-1}} N + \log T) \})$ regret.
