Active Learning for Non-Parametric Choice Models

We study the problem of actively learning a non-parametric choice model based on consumers' decisions. We present a negative result showing that such choice models may not be identifiable. To overcome the identifiability problem, we introduce a directed acyclic graph (DAG) representation of the choice model, which in a sense captures as much information about the choice model as could information-theoretically be identified. We then consider the problem of learning an approximation to this DAG representation in an active-learning setting. We design an efficient active-learning algorithm to estimate the DAG representation of the non-parametric choice model, which runs in polynomial time when the set of frequent rankings is drawn uniformly at random. Our algorithm learns the distribution over the most popular items of frequent preferences by actively and repeatedly offering assortments of items and observing the item chosen. We show that our algorithm can better recover a set of frequent preferences on both a synthetic and publicly available dataset on consumers' preferences, compared to the corresponding non-active learning estimation algorithms. This demonstrates the value of our algorithm and active-learning approaches more generally.