  We investigate contextual bandits in the presence of side-observations across
arms in order to design recommendation algorithms for users connected via
social networks. Users in social networks respond to their friends' activity,
and hence provide information about each other's preferences. In our model,
when a learning algorithm recommends an article to a user, not only does it
observe his/her response (e.g. an ad click), but also the side-observations,
i.e., the response of his neighbors if they were presented with the same
article. We model these observation dependencies by a graph $\mathcal{G}$ in
which nodes correspond to users, and edges correspond to social links. We
derive a problem/instance-dependent lower-bound on the regret of any consistent
algorithm. We propose an optimization (linear programming) based data-driven
learning algorithm that utilizes the structure of $\mathcal{G}$ in order to
make recommendations to users and show that it is asymptotically optimal, in
the sense that its regret matches the lower-bound as the number of rounds
$T\to\infty$. We show that this asymptotically optimal regret is upper-bounded
as $O\left(|\chi(\mathcal{G})|\log T\right)$, where $|\chi(\mathcal{G})|$ is
the domination number of $\mathcal{G}$. In contrast, a naive application of the
existing learning algorithms results in $O\left(N\log T\right)$ regret, where
$N$ is the number of users.
