The price of bandit information in multiclass online classification
This resolves open questions in online learning theory by providing tight bounds on the price of bandit information, which is crucial for applications like recommendation systems with limited feedback.
The paper tackles the problem of quantifying the cost of limited feedback in multiclass online classification, showing that the error rate ratio between full-information and bandit scenarios is at most 8·|Y|·log(|Y|) in the realizable case and Õ(√|Y|) in the agnostic case, with tight results up to logarithmic factors.
We consider two scenarios of multiclass online learning of a hypothesis class $H\subseteq Y^X$. 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 $8\cdot|Y|\cdot \log(|Y|)$ in the realizable case, and $\tilde{O}(\sqrt{|Y|})$ 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 $\reals^d$. We show that the bandit error rate of this class is $\tildeΘ(\frac{|Y|}{γ^2})$ in the realizable case and $\tildeΘ(\frac{1}γ\sqrt{|Y|T})$ in the agnostic case. This resolves an open question from (Kakade et. al. - Efficient bandit algorithms for online multiclass prediction).