  Generalized Linear Bandits (GLBs) are powerful extensions to the Linear
Bandit (LB) setting, broadening the benefits of reward parametrization beyond
linearity. In this paper we study GLBs in non-stationary environments,
characterized by a general metric of non-stationarity known as the
variation-budget or \emph{parameter-drift}, denoted $B_T$. While previous
attempts have been made to extend LB algorithms to this setting, they overlook
a salient feature of GLBs which flaws their results. In this work, we introduce
a new algorithm that addresses this difficulty. We prove that under a geometric
assumption on the action set, our approach enjoys a
$\tilde{\mathcal{O}}(B_T^{1/3}T^{2/3})$ regret bound. In the general case, we
show that it suffers at most a $\tilde{\mathcal{O}}(B_T^{1/5}T^{4/5})$ regret.
At the core of our contribution is a generalization of the projection step
introduced in Filippi et al. (2010), adapted to the non-stationary nature of
the problem. Our analysis sheds light on central mechanisms inherited from the
setting by explicitly splitting the treatment of the learning and tracking
aspects of the problem.
