  We propose improved fixed-design confidence bounds for the linear logistic
model. Our bounds significantly improve upon the state-of-the-art bound by Li
et al. (2017) via recent developments of the self-concordant analysis of the
logistic loss (Faury et al., 2020). Specifically, our confidence bound avoids a
direct dependence on $1/\kappa$, where $\kappa$ is the minimal variance over
all arms' reward distributions. In general, $1/\kappa$ scales exponentially
with the norm of the unknown linear parameter $\theta^*$. Instead of relying on
this worst-case quantity, our confidence bound for the reward of any given arm
depends directly on the variance of that arm's reward distribution. We present
two applications of our novel bounds to pure exploration and regret
minimization logistic bandits improving upon state-of-the-art performance
guarantees. For pure exploration, we also provide a lower bound highlighting a
dependence on $1/\kappa$ for a family of instances.
