  We study the problem of online multiclass classification in a setting where
the learner's feedback is determined by an arbitrary directed graph. While
including bandit feedback as a special case, feedback graphs allow a much
richer set of applications, including filtering and label efficient
classification. We introduce Gappletron, the first online multiclass algorithm
that works with arbitrary feedback graphs. For this new algorithm, we prove
surrogate regret bounds that hold, both in expectation and with high
probability, for a large class of surrogate losses. Our bounds are of order
$B\sqrt{\rho KT}$, where $B$ is the diameter of the prediction space, $K$ is
the number of classes, $T$ is the time horizon, and $\rho$ is the domination
number (a graph-theoretic parameter affecting the amount of exploration). In
the full information case, we show that Gappletron achieves a constant
surrogate regret of order $B^2K$. We also prove a general lower bound of order
$\max\big\{B^2K,\sqrt{T}\big\}$ showing that our upper bounds are not
significantly improvable. Experiments on synthetic data show that for various
feedback graphs, our algorithm is competitive against known baselines.
