  Logistic Bandits have recently attracted substantial attention, by providing
an uncluttered yet challenging framework for understanding the impact of
non-linearity in parametrized bandits. It was shown by Faury et al. (2020) that
the learning-theoretic difficulties of Logistic Bandits can be embodied by a
large (sometimes prohibitively) problem-dependent constant $\kappa$,
characterizing the magnitude of the reward's non-linearity. In this paper we
introduce a novel algorithm for which we provide a refined analysis. This
allows for a better characterization of the effect of non-linearity and yields
improved problem-dependent guarantees. In most favorable cases this leads to a
regret upper-bound scaling as $\tilde{\mathcal{O}}(d\sqrt{T/\kappa})$, which
dramatically improves over the $\tilde{\mathcal{O}}(d\sqrt{T}+\kappa)$
state-of-the-art guarantees. We prove that this rate is minimax-optimal by
deriving a $\Omega(d\sqrt{T/\kappa})$ problem-dependent lower-bound. Our
analysis identifies two regimes (permanent and transitory) of the regret, which
ultimately re-conciliates Faury et al. (2020) with the Bayesian approach of
Dong et al. (2019). In contrast to previous works, we find that in the
permanent regime non-linearity can dramatically ease the
exploration-exploitation trade-off. While it also impacts the length of the
transitory phase in a problem-dependent fashion, we show that this impact is
mild in most reasonable configurations.
