  We present a unified likelihood ratio-based confidence sequence (CS) for any
(self-concordant) generalized linear models (GLMs) 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 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)). Our analysis shows that the celebrated
optimistic approach simultaneously attains state-of-the-art regrets for various
self-concordant (not necessarily bounded) GLBs, and even poly(S)-free for
bounded GLBs, including logistic bandits. The regret analysis, our second
technical novelty, follows from combining our new CS with a new proof technique
that completely avoids the previously widely used self-concordant control lemma
(Faury et al., 2020, Lemma 9). Finally, we verify numerically that OFUGLB
significantly outperforms the prior state-of-the-art (Lee et al., 2024) for
logistic bandits.
