  We study a new non-stochastic federated multi-armed bandit problem with
multiple agents collaborating via a communication network. The losses of the
arms are assigned by an oblivious adversary that specifies the loss of each arm
not only for each time step but also for each agent, which we call ``doubly
adversarial". In this setting, different agents may choose the same arm in the
same time step but observe different feedback. The goal of each agent is to
find a globally best arm in hindsight that has the lowest cumulative loss
averaged over all agents, which necessities the communication among agents. We
provide regret lower bounds for any federated bandit algorithm under different
settings, when agents have access to full-information feedback, or the bandit
feedback. For the bandit feedback setting, we propose a near-optimal federated
bandit algorithm called FEDEXP3. Our algorithm gives a positive answer to an
open question proposed in Cesa-Bianchi et al. (2016): FEDEXP3 can guarantee a
sub-linear regret without exchanging sequences of selected arm identities or
loss sequences among agents. We also provide numerical evaluations of our
algorithm to validate our theoretical results and demonstrate its effectiveness
on synthetic and real-world datasets
