  We study the dynamic assortment planning problem, where for each arriving
customer, the seller offers an assortment of substitutable products and
customer makes the purchase among offered products according to an
uncapacitated multinomial logit (MNL) model. Since all the utility parameters
of MNL are unknown, the seller needs to simultaneously learn customers' choice
behavior and make dynamic decisions on assortments based on the current
knowledge. The goal of the seller is to maximize the expected revenue, or
equivalently, to minimize the expected regret. Although dynamic assortment
planning problem has received an increasing attention in revenue management,
most existing policies require the estimation of mean utility for each product
and the final regret usually involves the number of products $N$. The optimal
regret of the dynamic assortment planning problem under the most basic and
popular choice model---MNL model is still open. By carefully analyzing a
revenue potential function, we develop a trisection based policy combined with
adaptive confidence bound construction, which achieves an {item-independent}
regret bound of $O(\sqrt{T})$, where $T$ is the length of selling horizon. We
further establish the matching lower bound result to show the optimality of our
policy. There are two major advantages of the proposed policy. First, the
regret of all our policies has no dependence on $N$. Second, our policies are
almost assumption free: there is no assumption on mean utility nor any
"separability" condition on the expected revenues for different assortments.
Our result also extends the unimodal bandit literature.
