A note on a tight lower bound for capacitated 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.