  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 $O(\sqrt{K})$ gap
between upper and lower regret bounds from existing works.
