A Unified Confidence Sequence for Generalized Linear Models, with Applications to Bandits

We present a unified likelihood ratio-based confidence sequence (CS) for *any* (self-concordant) generalized linear model (GLM) that is guaranteed to be convex and numerically tight. We show that this is on par or improves upon known CSs for various GLMs, including Gaussian, Bernoulli, and Poisson. In particular, for the first time, our CS for Bernoulli has a \mathrm{poly}(S) -free radius where S is the norm of the unknown parameter. Our first technical novelty is its derivation, which utilizes a time-uniform PAC-Bayesian bound with a uniform prior/posterior, despite the latter being a rather unpopular choice for deriving CSs. As a direct application of our new CS, we propose a simple and natural optimistic algorithm called **OFUGLB**, applicable to *any* generalized linear bandits (**GLB**; Filippi et al. (2010)).