Beyond Bandit Feedback in Online Multiclass Classification

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 feed-
back 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√ρKT , where B is the diameter of the prediction space,
K is the number of classes, T is the time horizon, and ρ is the domination number
(a graph-theoretic parameter affecting the amount of exploration). In the full in-
formation case, we show that GAPPLETRON achieves a constant surrogate regret
of order B2K. We also prove a general lower bound of order max {B2K, √T }
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.