Robust Dynamic Assortment Optimization in the Presence of Outlier Customers

We consider the dynamic assortment optimization problem under the multinomial logit model (MNL) with unknown utility parameters. The main question investigated in this paper is model mis-specification under the $\varepsilon$-contamination model, which is a fundamental model in robust statistics and machine learning. In particular, throughout a selling horizon of length $T$, we assume that customers make purchases according to a well specified underlying multinomial logit choice model in a $(1-\varepsilon)$-fraction of the time periods, and make arbitrary purchasing decisions instead in the remaining $\varepsilon$-fraction of the time periods. In this model, we develop a new robust online assortment optimization policy via an active elimination strategy. We establish both upper and lower bounds on the regret, and show that our policy is optimal up to logarithmic factor in $T$ when the assortment capacity is constant. %% capacity of assortments has a constant upper limit. We further develop a fully adaptive policy that does not require any prior knowledge of the contamination parameter $\varepsilon$. In the case of the existence a sub-optimality gap between optimal and sub-optimal products, we also established gap-dependent logarithmic regret upper bounds and lower bounds in both the known-$\varepsilon$ and unknown-$\varepsilon$ cases. Our simulation study shows that our policy outperforms the existing policies based on upper confidence bounds (UCB) and Thompson sampling.