assortment optimization in an adversarial setting rather than stochastic setting. For example, [@Golrezaei2014; @Chen:16recom] assumed that each customer’s choice behavior is known, but that the customers’ arriving sequence (or customers’ types) can be adversarially chosen and took the inventory level into consideration. Since the arriving sequence can be arbitrary, there is no learning component in the problem and both [@Golrezaei2014] and [@Chen:16recom] adopted the competitive ratio as the performance evaluation metric.

Another field of related research is the *contextual bandit* literature, in which the linear contextual bandit has been widely studied as a special case (see, e.g., [@Dani08stochastic; @Rusmevichientong:10:bandit; @chu11; @Abbasi-Yadkori:2011; @Agrawal:13] and references therein). Some recent work extends the linear contextual bandit to *generalized linear bandit* [@filippi2010parametric; @li2017provably], which assumes a generalized linear reward structure. In particular, the reward $r$ of pulling an arm given the observed feature vector of this arm $x$ is modeled by $$\mathbb E[r|x] = \sigma(x^\top\theta_0),
\label{eq:glm}$$ for an unknown linear model $\theta_0$ and a known link function $\sigma:\mathbb R\to\mathbb R$. For example, for a linear contextual bandit, $\sigma$ is the identity mapping, i.e., $\mathbb E[r|x]=\sigma(x^\top\theta_0)$. For the logistic contextual bandit, we have $r \in \{0,1\}$ and $\Pr(r=1|x)=\frac{\exp(x^\top\theta_0)}{1+\exp(x^\top\theta_0)}$. [ In a standard generalized linear bandit 