MNL-Bandit with Knapsacks

We consider a dynamic assortment selection problem where a seller has a fixed inventory of $N$ substitutable products and faces an unknown demand that arrives sequentially over $T$ periods. In each period, the seller needs to decide on the assortment of products (of cardinality at most $K$) to offer to the customers. The customer's response follows an unknown multinomial logit model (MNL) with parameters $v$. The goal of the seller is to maximize the total expected revenue given the fixed initial inventory of $N$ products. We give a policy that achieves a regret of $\tilde O\Big(K \sqrt{KN T}\Big(\sqrt{v_{\text{max}}} + \frac{1}{q_{\text{min}}}\text{OPT}\Big)\Big)$, where $v_{\text{max}}\leq 1$ is the maximum utility for any product and $q_{\text{min}}$ the minimum inventory level, under a mild assumption on the model parameters. In particular, our policy achieves a near-optimal $\tilde O(\sqrt{T})$ regret in a large-inventory setting. Our policy builds upon the UCB-based approach for MNL-bandit without inventory constraints in [1] and addresses the inventory constraints through an exponentially sized LP for which we present a tractable approximation while keeping the $\tilde O(\sqrt{T})$ regret bound.