Approximating a RUM from Distributions on $k$-Slates

In this work we consider the problem of fitting Random Utility Models (RUMs) to user choices. Given the winner distributions of the subsets of size $k$ of a universe, we obtain a polynomial-time algorithm that finds the RUM that best approximates the given distribution on average. Our algorithm is based on a linear program that we solve using the ellipsoid method. Given that its separation oracle problem is NP-hard, we devise an approximate separation oracle that can be viewed as a generalization of the weighted Feedback Arc Set problem to hypergraphs. Our theoretical result can also be made practical: we obtain a heuristic that scales to real-world datasets.