  We consider a dynamic assortment selection problem where a seller has a fixed
inventory of $N$ substitutable products and faces an unknown demand that
arrives sequentially over $T$ periods. In each period, the seller needs to
decide on the assortment of products (satisfying certain constraints) to offer
to the customers. The customer's response follows an unknown multinomial logit
model (MNL) with parameter $\boldsymbol{v}$. If customer selects product $i \in
[N]$, the seller receives revenue $r_i$. The goal of the seller is to maximize
the total expected revenue from the $T$ customers given the fixed initial
inventory of $N$ products. We present MNLwK-UCB, a UCB-based algorithm and
characterize its regret under different regimes of inventory size. We show that
when the inventory size grows quasi-linearly in time, MNLwK-UCB achieves a
$\tilde{O}(N + \sqrt{NT})$ regret bound. We also show that for a smaller
inventory (with growth $\sim T^{\alpha}$, $\alpha < 1$), MNLwK-UCB achieves a
$\tilde{O}(N(1 + T^{\frac{1 - \alpha}{2}}) + \sqrt{NT})$. In particular, over a
long time horizon $T$, the rate $\tilde{O}(\sqrt{NT})$ is always achieved
regardless of the constraints and the size of the inventory.
