0) \in \mathbb{R}^{DN}$, which is a concatenation of $N$ $D$-dimensional vectors with the $j$-th vector being $x_t$ and all other vectors being 0. Then according to our linear model in Eq. , we have $u_{tj}= v_{tj}^\top\theta_0= x_t^\top \theta(j)$, which recovers the personalized MNL model. Using our regret bound ${\widetilde}{O}(d\sqrt{T})$ with $d=DN$ as the dimensionality of $\theta_0$, we directly obtain the regret ${\widetilde}{O}(DN \sqrt{T})$ for the dynamic assortment optimization under the personalized MNL. As compared to the Bayesian regret bound ${\widetilde}{O}(DN\sqrt{KT})$ in [@Wang:17:person] (see Theorem 3.3. therein), our approach still saves a factor of $\sqrt{K}$.

We also remark that our results require a slightly stronger assumption on the contextual information vectors $\{v_{tj}\}$ compared to [@Wang:17:person], which allows customer feature vectors $\{x_t\}$ to be adversarially chosen. More specifically, [a stochastic assumption]{} is imposed on $\{v_{tj}\}$ only during the *pure exploration* phase of our proposed policy. [After this pure exploration phase, the feature vectors $\{v_{tj}\}$ can also be adversarially chosen.]{} We refer the readers to Sec. \[subsec:regret\] for further details.

In addition, the developed techniques in our work and [@Wang:17:person] are different, Our policy is based on UCB, while the policy in [@Wang:17:person] is based on Thompson sampling. Furthermore, other research studies personalized 