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Logistic bandit is a ubiquitous framework of modeling users' choices, e.g.,click vs. no click for advertisement recommender system. We observe that theprior works overlook or neglect dependencies in $S \geq \lVert \theta_\star\rVert_2$, where $\theta_\star \in \mathbb{R}^d$ is the unknown parametervector, which is particularly problematic when $S$ is large, e.g., $S \geq d$.In this work, we improve the dependency on $S$ via a novel approach called {\itregret-to-confidence set conversion (R2CS)}, which allows us to construct aconvex confidence set based on only the \textit{existence} of an onlinelearning algorithm with a regret guarantee. Using R2CS, we obtain a strictimprovement in the regret bound w.r.t. $S$ in logistic bandits while retainingcomputational feasibility and the dependence on other factors such as $d$ and$T$. We apply our new confidence set to the regret analyses of logistic banditswith a new martingale concentration step that circumvents an additional factorof $S$. We then extend this analysis to multinomial logistic bandits and obtainsimilar improvements in the regret, showing the efficacy of R2CS. While weapplied R2CS to the (multinomial) logistic model, R2CS is a generic approachfor developing confidence sets that can be used for various models, which canbe of independent interest.

Improved Regret Bounds of (Multinomial) Logistic Bandits via Regret-to-Confidence-Set Conversion