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

Junghyun Lee, Se-Young Yun, Kwang-Sung Jun

Keywords: generalized linear model, bandit, UCB, anytime-valid, time-uniform, confidence sequence, PAC-Bayes

Abstract: 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 $\mathrm{poly}(S)$-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)). Our analysis shows that the celebrated optimistic approach attains the state-of-the-art regrets for various self-concordant (not necessarily bounded) **GLB**s, and even $\mathrm{poly}(S)$-free for bounded **GLB**s, 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), which may be of independent interest. Finally, we verify numerically that **OFUGLB** significantly outperforms the prior state-of-the-art (Lee et al., 2024) for logistic bandits.

Submission Number: 57