Abstract: The modern retailing system is witnessing fast updating in product features and customer behaviors, entailing adaptive policies that can effectively capture the dynamics of customer preferences. To optimize potential revenues and manage the risks associated with changing
customer preferences, it is important to develop an online framework that quantifies the uncertainty of the optimal assortment adaptively.

We study the combinatorial inference of the optimal assortment within the framework of the contextual multinomial logit model. In this setting, customer choice outcomes are actively collected over a series of time points, where the contextual information for products—including embedding vectors that capture the customer-product dynamics, as well as revenue parameters—varies over time. Using a dynamic policy, the offer set is adaptively selected at each time point based on historical data. We propose an inferential procedure that constructs a discrete confidence set for the true optimal assortment based upon the data collected by the dynamic policy, which can be applied to test any combinatorial properties of the optimal assortment, such as the number of product categories to include in the offer set.

The temporal dependency and combinatorial data structure due to adaptive sampling create challenges for convergence analysis. To address these, we develop new probabilistic results on anti-concentration bounds for the difference between the maxima of two Gaussian random vectors. Furthermore, we address the high dimensionality of the combinatorial inference problem by employing discretization via epsilon-covering and subspace projection techniques. We provide theoretical guarantees on both the validity and power of our inferential procedure, and establish information-theoretic lower bounds for the required signal strength, which match the upper bounds of our procedure up to logarithmic factors.

https://judygiant.github.io/