  In this work we consider the problem of regret minimization for logistic
bandits. The main challenge of logistic bandits is reducing the dependence on a
potentially large problem dependent constant $\kappa$ that can at worst scale
exponentially with the norm of the unknown parameter $\theta_{\ast}$. Abeille
et al. (2021) have applied self-concordance of the logistic function to remove
this worst-case dependence providing regret guarantees like
$O(d\log^2(\kappa)\sqrt{\dot\mu T}\log(|\mathcal{X}|))$ where $d$ is the
dimensionality, $T$ is the time horizon, and $\dot\mu$ is the variance of the
best-arm. This work improves upon this bound in the fixed arm setting by
employing an experimental design procedure that achieves a minimax regret of
$O(\sqrt{d \dot\mu T\log(|\mathcal{X}|)})$. Our regret bound in fact takes a
tighter instance (i.e., gap) dependent regret bound for the first time in
logistic bandits. We also propose a new warmup sampling algorithm that can
dramatically reduce the lower order term in the regret in general and prove
that it can replace the lower order term dependency on $\kappa$ to
$\log^2(\kappa)$ for some instances. Finally, we discuss the impact of the bias
of the MLE on the logistic bandit problem, providing an example where $d^2$
lower order regret (cf., it is $d$ for linear bandits) may not be improved as
long as the MLE is used and how bias-corrected estimators may be used to make
it closer to $d$.
