  We consider a dynamic assortment selection problem, where in every round the
retailer offers a subset (assortment) of $N$ substitutable products to a
consumer, who selects one of these products according to a multinomial logit
(MNL) choice model. The retailer observes this choice and the objective is to
dynamically learn the model parameters, while optimizing cumulative revenues
over a selling horizon of length $T$. We refer to this exploration-exploitation
formulation as the MNL-Bandit problem. Existing methods for this problem follow
an "explore-then-exploit" approach, which estimate parameters to a desired
accuracy and then, treating these estimates as if they are the correct
parameter values, offers the optimal assortment based on these estimates. These
approaches require certain a priori knowledge of "separability", determined by
the true parameters of the underlying MNL model, and this in turn is critical
in determining the length of the exploration period. (Separability refers to
the distinguishability of the true optimal assortment from the other
sub-optimal alternatives.) In this paper, we give an efficient algorithm that
simultaneously explores and exploits, achieving performance independent of the
underlying parameters. The algorithm can be implemented in a fully online
manner, without knowledge of the horizon length $T$. Furthermore, the algorithm
is adaptive in the sense that its performance is near-optimal in both the "well
separated" case, as well as the general parameter setting where this separation
need not hold.
