supplementary material.

The problem setup {#sec:model}
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There are $N$ items, conveniently labeled as $1,2,\cdots,N$. At each time $t$, a set of time-sensitive “feature vectors” $v_{t1}, v_{t2}, \cdots, v_{tN}\in\mathbb R^d$ and revenues $r_{t1},\cdots,r_{tN}\in[0,1]$ are observed, reflecting time-varying changes of items’ revenues and customers’ preferences. A retailer, based on the features $\{v_{ti}\}_{i=1}^N$ and previous purchasing actions, picks an assortment $S_t\subseteq[N]$ under the cardinality constraint $|S_t|\leq K$ to present to an incoming customer; the retailer then observes a purchasing action $i_t\in S_t\cup\{0\}$ and collects the associated revenue $r_{i_t}$ of the purchased item (if $i_t=0$ then no item is purchased and zero revenue is collected).

We use an MNL model with features to characterize how a customer makes choices. Let $\theta_0\in\mathbb R^d$ be an *unknown* time-invariant coefficient. For any $S\subseteq[N]$, the choice model $p_{\theta_0,t}(\cdot|S)$ is specified as (let $r_0=0$ and $v_{t0}=0$) $$p_{\theta_0,t}(j|S) = \frac{\exp\{v_{tj}^\top\theta_0\}}{1+\sum_{k\in S} \exp\{v_{tk}^\top\theta_0\}} \;\;\;\;\;\;\forall j\in S\cup\{0\}.
\label{eq:context_MNL}$$ For simplicity, in the rest of the paper we use $p_{\theta,t}(\cdot|S)$ to denote the law of the purchased item $i_t$ conditioned on given assortment $S$ at time $t$, parameterized by the coefficient $\theta\in\mathbb R^d$. The expected revenue $R_t(S)$ of assortment $S\subseteq[N]$ at time $t$ is then given by $$\label{eq:rev}
R_t(S) := \mathbb E_{\theta_0,t}[r_{tj}|S] 