  We study the multinomial logit (MNL) bandit problem, where at each time step,
the seller offers an assortment of size at most $K$ from a pool of $N$ items,
and the buyer purchases an item from the assortment according to a MNL choice
model. The objective is to learn the model parameters and maximize the expected
revenue. We present (i) an algorithm that identifies the optimal assortment
$S^*$ within $\widetilde{O}(\sum_{i = 1}^N \Delta_i^{-2})$ time steps with high
probability, and (ii) an algorithm that incurs $O(\sum_{i \notin S^*}
K\Delta_i^{-1} \log T)$ regret in $T$ time steps. To our knowledge, our
algorithms are the first to achieve gap-dependent bounds that fully depends on
the suboptimality gaps of all items. Our technical contributions include an
algorithmic framework that relates the MNL-bandit problem to a variant of the
top-$K$ arm identification problem in multi-armed bandits, a generalized
epoch-based offering procedure, and a layer-based adaptive estimation
procedure.
