the unknown coefficient $\theta_0$, i.e., a good initial estimator for $\theta_0$. For the simplicity of the analysis, we choose one item for each assortment in this phase, which facilitates us to adapt existing analysis in [@filippi2010parametric; @li2017provably] as the MNL-logit choice model reduces to a *generalized linear model* when only one item is present in the assortment. In the second phase, we use a UCB-type approach that selects $S_t$ as the assortment maximizing an *upper bound* $\overline R_t(S_t)$ of the expected revenue $R_t(S_t)$. Such upper bounds are built using a *local Maximum Likelihood Estimation (MLE)* of $\theta_0$. In particular, in Step 5, instead of computing an MLE, we compute a local MLE, where the estimator ${\widehat}{\theta}_{t-1}$ lies in a ball centered at the pilot estimator $\theta^*$ with a radius $\tau$. This localization also simplifies the technical analysis based on Taylor expansion, which benefits from the constraint that ${\widehat}\theta_{t-1}$ is not too far away from $\theta^*$.

To construct the confidence bound, we introduce the matrices ${\widehat}M_t({\widehat}\theta_{t-1}|S)$ and ${\widehat}I_{t-1}({\widehat}\theta_{t-1})$ in Step 6 of Algorithm \[alg:mle-ucb\], which are empirical estimates of the *Fisher’s information* matrices $-\mathbb E[\nabla_\theta^2\log p(\cdot|\theta)]$ corresponding to the MNL choice model $p(\cdot|S_t)$. The population version of the Fisher’s information matrices 