  In this paper, we study the contextual multinomial logit (MNL) bandit problem
in which a learning agent sequentially selects an assortment based on
contextual information, and user feedback follows an MNL choice model. There
has been a significant discrepancy between lower and upper regret bounds,
particularly regarding the maximum assortment size $K$. Additionally, the
variation in reward structures between these bounds complicates the quest for
optimality. Under uniform rewards, where all items have the same expected
reward, we establish a regret lower bound of $\Omega(d\sqrt{\smash[b]{T/K}})$
and propose a constant-time algorithm, OFU-MNL+, that achieves a matching upper
bound of $\tilde{O}(d\sqrt{\smash[b]{T/K}})$. Under non-uniform rewards, we
prove a lower bound of $\Omega(d\sqrt{T})$ and an upper bound of
$\tilde{O}(d\sqrt{T})$, also achievable by OFU-MNL+. Our empirical studies
support these theoretical findings. To the best of our knowledge, this is the
first work in the contextual MNL bandit literature to prove minimax optimality
-- for either uniform or non-uniform reward setting -- and to propose a
computationally efficient algorithm that achieves this optimality up to
logarithmic factors.
