Provably Optimal Algorithms for Generalized Linear Contextual Bandits

TL;DR

This paper introduces a provably optimal algorithm for generalized linear contextual bandits, achieving minimax regret bounds and providing new confidence bounds for maximum-likelihood estimates, advancing theoretical understanding in this area.

Contribution

It proposes the first UCB-based algorithm for generalized linear bandits with optimal regret bounds and introduces a novel finite-sample confidence bound for MLEs in GLMs.

Findings

Achieves $ ilde{O}( oot{d}T)$ regret matching the lower bound

Improves previous results by a $ oot{d}$ factor for fixed arms

Provides a new confidence bound for MLEs in GLMs

Abstract

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.