the third step that an upper bound on the estimation error ${\widehat}\theta_{t-1}-\theta_0$ implies an upper bound on the estimation error of the expected revenue $R_t(S)$, hence showing that ${\overline}R_t(S)$ are valid upper confidence bounds. Finally, we apply the *elliptical potential lemma*, which also plays a key role in linear stochastic bandit and its variants, to complete our proof.

### Analysis of pure exploration and the pilot estimator

Our first step is to establish an upper bound on the estimation error $\|\theta^*-\theta_0\|_2$ of the pilot estimator $\theta^*$, built using pure exploration data. It should be noted that in the pure exploration phase ($t\in\{1,\cdots,T_0\}$), the assortments $\{S_t\}_{t=1}^{T_0}$ only consist of one item. Therefore the observation model reduces to a standard *generalized linear model* with the sigmoid function $\sigma(x)=1/(1+e^{-x})= e^x/(1+e^x)$ as the link function, which is essentially a logistic regression model of observing 1 if the customer makes a purchase. Because the choice model in the pure exploration phase reduces to a generalized linear model, we can cite existing works to upper bound the error $\|\theta^*-\theta_0\|_2$. In particular, the following lemma is cited from [@li2017provably Eq. (18)], adapted to our model and parameter settings. The details on how to adapt the result from 