  Logistic bandit is a ubiquitous framework of modeling users' choices, e.g.,
click vs. no click for advertisement recommender system. We observe that the
prior works overlook or neglect dependencies in $S \geq \lVert \theta_\star
\rVert_2$, where $\theta_\star \in \mathbb{R}^d$ is the unknown parameter
vector, 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 {\it
regret-to-confidence set conversion (R2CS)}, which allows us to construct a
convex confidence set based on only the \textit{existence} of an online
learning algorithm with a regret guarantee. Using R2CS, we obtain a strict
improvement in the regret bound w.r.t. $S$ in logistic bandits while retaining
computational feasibility and the dependence on other factors such as $d$ and
$T$. We apply our new confidence set to the regret analyses of logistic bandits
with a new martingale concentration step that circumvents an additional factor
of $S$. We then extend this analysis to multinomial logistic bandits and obtain
similar improvements in the regret, showing the efficacy of R2CS. While we
applied R2CS to the (multinomial) logistic model, R2CS is a generic approach
for developing confidence sets that can be used for various models, which can
be of independent interest.
