  In this paper, we consider the contextual variant of the MNL-Bandit problem.
More specifically, we consider a dynamic set optimization problem, where a
decision-maker offers a subset (assortment) of products to a consumer and
observes the response in every round. Consumers purchase products to maximize
their utility. We assume that a set of attributes describe the products, and
the mean utility of a product is linear in the values of these attributes. We
model consumer choice behavior using the widely used Multinomial Logit (MNL)
model and consider the decision maker problem of dynamically learning the model
parameters while optimizing cumulative revenue over the selling horizon $T$.
Though this problem has attracted considerable attention in recent times, many
existing methods often involve solving an intractable non-convex optimization
problem. Their theoretical performance guarantees depend on a problem-dependent
parameter which could be prohibitively large. In particular, existing
algorithms for this problem have regret bounded by $O(\sqrt{\kappa d T})$,
where $\kappa$ is a problem-dependent constant that can have an exponential
dependency on the number of attributes. In this paper, we propose an optimistic
algorithm and show that the regret is bounded by $O(\sqrt{dT} + \kappa)$,
significantly improving the performance over existing methods. Further, we
propose a convex relaxation of the optimization step, which allows for
tractable decision-making while retaining the favourable regret guarantee.
