A Note on a Tight Lower Bound for MNL-Bandit Assortment Selection Models

In this short note we consider a dynamic assortment planning problem under the capacitated multinomial logit (MNL) bandit model. We prove a tight lower bound on the accumulated regret that matches existing regret upper bounds for all parameters (time horizon T, number of items N and maximum assortment capacity K) up to logarithmic factors. Our results close an O(√K) gap between upper and lower regret bounds from existing works.