  We present Gaptron, a randomized first-order algorithm for online multiclass
classification. In the full information setting we show expected mistake bounds
with respect to the logistic loss, hinge loss, and the smooth hinge loss with
constant regret, where the expectation is with respect to the learner's
randomness. In the bandit classification setting we show that Gaptron is the
first linear time algorithm with $O(K\sqrt{T})$ expected regret, where $K$ is
the number of classes. Additionally, the expected mistake bound of Gaptron does
not depend on the dimension of the feature vector, contrary to previous
algorithms with $O(K\sqrt{T})$ regret in the bandit classification setting. We
present a new proof technique that exploits the gap between the zero-one loss
and surrogate losses rather than exploiting properties such as exp-concavity or
mixability, which are traditionally used to prove logarithmic or constant
regret bounds.
