the contextual information. [@Chen:18dynamic] further studied the dynamic assortment optimization under nested logit models. We also note that to highlight our key idea and focus on the balance between learning of $\theta_0$ and revenue maximization, we study the stylized dynamic assortment optimization problems following the existing literature [@Rusmevichientong2010; @Saure2013; @Agrawal16MNLBandit; @Agrawal17Thompson], which ignore operations considerations such as price decisions and inventory replenishment.

There is another line of recent research on investigating personalized assortment optimization.

By incorporating the feature information of each arriving customer, both the static and dynamic assortment optimization problems are studied in [@Chen2015] and [@Wang:17:person], respectively. It is worthwhile noting that although we do not approach our work from a personalized perspective[^3], the *personalized MNL* considered in [@Wang:17:person] can be viewed as a special of our model.

In particular, the personalized MNL assumes that each product $j$ is associated with an unknown coefficient $\theta(j) \in \mathbb{R}^D$. When a customer arrives at time $t$ with the *observed feature $x_t$*, the utility of product $j$ at time $t$ is $u_{tj}=x_t^\top \theta(j)$. Now we explain how to specialize our model to obtain the personalized MNL. Let us define $\theta_0:=\{\theta(1), \ldots, \theta(N)\} \in \mathbb{R}^{DN}$ and the feature vector $v_{tj}:=(0, \ldots, x_{t}, \ldots, 