Learning an arbitrary mixture of two multinomial logits
This work addresses a foundational limitation in choice modeling for fields like economics and marketing, though it is incremental as it extends prior results from uniform to arbitrary mixtures.
The paper tackles the problem of learning an arbitrary mixture of two multinomial logit models, which can approximate any random utility model, by showing that identifiability fails only on a negligible measure and devising an algorithm with polynomial sample and linear query complexity for identifiable cases.
In this paper, we consider mixtures of multinomial logistic models (MNL), which are known to $ε$-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.