  We consider a sequential assortment selection problem where the user choice
is given by a multinomial logit (MNL) choice model whose parameters are
unknown. In each period, the learning agent observes a $d$-dimensional
contextual information about the user and the $N$ available items, and offers
an assortment of size $K$ to the user, and observes the bandit feedback of the
item chosen from the assortment. We propose upper confidence bound based
algorithms for this MNL contextual bandit. The first algorithm is a simple and
practical method which achieves an $\tilde{\mathcal{O}}(d\sqrt{T})$ regret over
$T$ rounds. Next, we propose a second algorithm which achieves a
$\tilde{\mathcal{O}}(\sqrt{dT})$ regret. This matches the lower bound for the
MNL bandit problem, up to logarithmic terms, and improves on the best known
result by a $\sqrt{d}$ factor. To establish this sharper regret bound, we
present a non-asymptotic confidence bound for the maximum likelihood estimator
of the MNL model that may be of independent interest as its own theoretical
contribution. We then revisit the simpler, significantly more practical, first
algorithm and show that a simple variant of the algorithm achieves the optimal
regret for a broad class of important applications.
