  This paper studies the adversarial graphical contextual bandits, a variant of
adversarial multi-armed bandits that leverage two categories of the most common
side information: \emph{contexts} and \emph{side observations}. In this
setting, a learning agent repeatedly chooses from a set of $K$ actions after
being presented with a $d$-dimensional context vector. The agent not only
incurs and observes the loss of the chosen action, but also observes the losses
of its neighboring actions in the observation structures, which are encoded as
a series of feedback graphs. This setting models a variety of applications in
social networks, where both contexts and graph-structured side observations are
available. Two efficient algorithms are developed based on \texttt{EXP3}. Under
mild conditions, our analysis shows that for undirected feedback graphs the
first algorithm, \texttt{EXP3-LGC-U}, achieves the regret of order
$\mathcal{O}(\sqrt{(K+\alpha(G)d)T\log{K}})$ over the time horizon $T$, where
$\alpha(G)$ is the average \emph{independence number} of the feedback graphs. A
slightly weaker result is presented for the directed graph setting as well. The
second algorithm, \texttt{EXP3-LGC-IX}, is developed for a special class of
problems, for which the regret is reduced to
$\mathcal{O}(\sqrt{\alpha(G)dT\log{K}\log(KT)})$ for both directed as well as
undirected feedback graphs. Numerical tests corroborate the efficiency of
proposed algorithms.
