  In this paper, we study the MNL-Bandit problem in a non-stationary
environment and present an algorithm with a worst-case expected regret of
$\tilde{O}\left( \min \left\{ \sqrt{NTL}\;,\;
N^{\frac{1}{3}}(\Delta_{\infty}^{K})^{\frac{1}{3}} T^{\frac{2}{3}} +
\sqrt{NT}\right\}\right)$. Here $N$ is the number of arms, $L$ is the number of
changes and $\Delta_{\infty}^{K}$ is a variation measure of the unknown
parameters. Furthermore, we show matching lower bounds on the expected regret
(up to logarithmic factors), implying that our algorithm is optimal. Our
approach builds upon the epoch-based algorithm for stationary MNL-Bandit in
Agrawal et al. 2016. However, non-stationarity poses several challenges and we
introduce new techniques and ideas to address these. In particular, we give a
tight characterization for the bias introduced in the estimators due to non
stationarity and derive new concentration bounds.
