The price of bandit information in multiclass online classification

We consider two scenarios of multiclass online learning of a hypothesis class Hsubseteq YX. In the em full information scenario, the learner is exposed to instances together with their labels. In the em bandit scenario, the true label is not exposed, but rather an indication whether the learner's prediction is correct or not. We show that the ratio between the error rates in the two scenarios is at most 8·|Y|· log(|Y|) in the realizable case, and tildeO(√|Y|) in the agnostic case. The results are tight up to a logarithmic factor and essentially answer an open question from (Daniely et. al. - Multiclass learnability and the erm principle). We apply these results to the class of γ-margin multiclass linear classifiers in realsd. We show that the bandit error rate of this class is tildeTheta(frac|Y|γ2) in the realizable case and tildeTheta(frac1γ√|Y|T) in the agnostic case. This resolves an open question from (Kakade et. al. - Efficient bandit algorithms for online multiclass prediction).