Learning an arbitrary mixture of two multinomial logits

TL;DR

This paper develops a theoretical framework and an algorithm for learning any mixture of two multinomial logistic models, extending previous results limited to uniform mixtures, with guarantees on identifiability and sample complexity.

Contribution

It proves that arbitrary mixtures of two MNL models are identifiable except on negligible cases and provides a polynomial-sample, linear-query algorithm for learning them.

Findings

Mixtures are identifiable except on a measure-zero set.

Proposed algorithm learns mixtures with polynomial samples.

Numerical experiments support theoretical results.

Abstract

In this paper, we consider mixtures of multinomial logistic models (MNL), which are known to $\epsilon$-approximate any random utility model. Despite its long history and broad use, rigorous results are only available for learning a uniform mixture of two MNLs. Continuing this line of research, we study the problem of learning an arbitrary mixture of two MNLs. We show that the identifiability of the mixture models may only fail on an algebraic variety of a negligible measure. This is done by reducing the problem of learning a mixture of two MNLs to the problem of solving a system of univariate quartic equations. We also devise an algorithm to learn any mixture of two MNLs using a polynomial number of samples and a linear number of queries, provided that a mixture of two MNLs over some finite universe is identifiable. Several numerical experiments and conjectures are also presented.