Provably Optimal Algorithms for Generalized Linear Contextual Bandits
This provides a provably optimal solution for contextual bandit problems with binary rewards, addressing a gap in theoretical analyses that previously focused on linear models.
The paper tackles the problem of designing algorithms for generalized linear contextual bandits, which are used in applications like news recommendation and advertising, and achieves an $ ilde{O}(\sqrt{dT})$ regret that matches the minimax lower bound, improving on the best previous result by a $\sqrt{d}$ factor.
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.