Efficiently Learning Mixtures of Mallows Models

TL;DR

This paper presents the first polynomial-time algorithm for learning parameters of mixtures of Mallows models with any constant number of components, advancing the understanding of ranking data modeling.

Contribution

It introduces a novel polynomial-time algorithm for learning mixtures of Mallows models with multiple components, using a determinantal identity and peeling technique.

Findings

Polynomial identifiability of mixture models

Sample complexity lower bounds established

Faster algorithms under parameter separation

Abstract

Mixtures of Mallows models are a popular generative model for ranking data coming from a heterogeneous population. They have a variety of applications including social choice, recommendation systems and natural language processing. Here we give the first polynomial time algorithm for provably learning the parameters of a mixture of Mallows models with any constant number of components. Prior to our work, only the two component case had been settled. Our analysis revolves around a determinantal identity of Zagier which was proven in the context of mathematical physics, which we use to show polynomial identifiability and ultimately to construct test functions to peel off one component at a time. To complement our upper bounds, we show information-theoretic lower bounds on the sample complexity as well as lower bounds against restricted families of algorithms that make only local queries. Together, these results demonstrate various impediments to improving the dependence on the number of components. They also motivate the study of learning mixtures of Mallows models from the perspective of beyond worst-case analysis. In this direction, we show that when the scaling parameters of the Mallows models have separation, there are much faster learning algorithms.