  We study agents communicating over an underlying network by exchanging
messages, in order to optimize their individual regret in a common
nonstochastic multi-armed bandit problem. We derive regret minimization
algorithms that guarantee for each agent $v$ an individual expected regret of
$\widetilde{O}\left(\sqrt{\left(1+\frac{K}{\left|\mathcal{N}\left(v\right)\right|}\right)T}\right)$,
where $T$ is the number of time steps, $K$ is the number of actions and
$\mathcal{N}\left(v\right)$ is the set of neighbors of agent $v$ in the
communication graph. We present algorithms both for the case that the
communication graph is known to all the agents, and for the case that the graph
is unknown. When the graph is unknown, each agent knows only the set of its
neighbors and an upper bound on the total number of agents. The individual
regret between the models differs only by a logarithmic factor. Our work
resolves an open problem from [Cesa-Bianchi et al., 2019b].
