Learning an arbitrary mixture of two multinomial logits

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 Lebesgue measure $0$, implying that all existing algorithms apply in the almost sure sense. 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. As a byproduct, we derive an algorithm to learn any mixture of two MNLs in linear time provided that a mixture of two MNLs over some finite universe is identifiable. Several numerical experiments and conjectures are also presented.