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

A Tractable Online Learning Algorithm for the Multinomial Logit Contextual Bandit