  Contextual bandits are widely used in Internet services from news
recommendation to advertising, and to Web search. Generalized linear models
(logistical regression in particular) have demonstrated stronger performance
than linear models in many applications where rewards are binary. However, most
theoretical analyses on contextual bandits so far are on linear bandits. In
this work, we propose an upper confidence bound based algorithm for generalized
linear contextual bandits, which achieves an $\tilde{O}(\sqrt{dT})$ regret over
$T$ rounds with $d$ dimensional feature vectors. This regret matches the
minimax lower bound, up to logarithmic terms, and improves on the best previous
result by a $\sqrt{d}$ factor, assuming the number of arms is fixed. A key
component in our analysis is to establish a new, sharp finite-sample confidence
bound for maximum-likelihood estimates in generalized linear models, which may
be of independent interest. We also analyze a simpler upper confidence bound
algorithm, which is useful in practice, and prove it to have optimal regret for
certain cases.
