Mallows Model with Learned Distance Metrics: Sampling and Maximum Likelihood Estimation

2025-07-10

AI Summary

Existing Mallows models rely on fixed distance metrics (e.g., Kendall’s τ), limiting their ability to capture heterogeneous ranking variations across domains such as sports rankings. This work introduces the first learnable-distance Mallows model: it parameterizes the distance function using the Lₐ family, jointly modeling the distance metric, central ranking, and dispersion parameter. We design an FPTAS sampling algorithm applicable to arbitrary α and β, enabling efficient sample-based maximum-likelihood estimation. Theoretical analysis establishes strong consistency of the resulting estimators. Empirical evaluation on real-world sports ranking data demonstrates significant improvements over fixed-metric baselines in both goodness-of-fit and predictive accuracy. Our approach establishes a new paradigm for context-adaptive ranking modeling by endowing the Mallows framework with data-driven metric learning.

Technology Category

• Machine Learning: Learning Preferences or Rankings
• Reasoning under Uncertainty: Relational Probabilistic Models
• Search and Optimization: Learning to Search

Application Category

• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

extit{Mallows model} is a widely-used probabilistic framework for learning from ranking data, with applications ranging from recommendation systems and voting to aligning language models with human preferences~cite{chen2024mallows, kleinberg2021algorithmic, rafailov2024direct}. Under this model, observed rankings are noisy perturbations of a central ranking $σ$, with likelihood decaying exponentially in distance from $σ$, i.e, $P (π) propto expig(-βcdot d(π, σ)ig),$ where $β> 0$ controls dispersion and $d$ is a distance function. Existing methods mainly focus on fixed distances (such as Kendall's $τ$ distance), with no principled approach to learning the distance metric directly from data. In practice, however, rankings naturally vary by context; for instance, in some sports we regularly see long-range swaps (a low-rank team beating a high-rank one), while in others such events are rare. Motivated by this, we propose a generalization of Mallows model that learns the distance metric directly from data. Specifically, we focus on $L_α$ distances: $d_α(π,σ):=sum_{i=1} |π(i)-σ(i)|^α$. For any $αgeq 1$ and $β>0$, we develop a Fully Polynomial-Time Approximation Scheme (FPTAS) to efficiently generate samples that are $ε$- close (in total variation distance) to the true distribution. Even in the special cases of $L_1$ and $L_2$, this generalizes prior results that required vanishing dispersion ($β o0$). Using this sampling algorithm, we propose an efficient Maximum Likelihood Estimation (MLE) algorithm that jointly estimates the central ranking, the dispersion parameter, and the optimal distance metric. We prove strong consistency results for our estimators (for any values of $α$ and $β$), and we validate our approach empirically using datasets from sports rankings.

Problem

• Learning distance metrics from ranking data in Mallows model
• Developing efficient sampling algorithm for Mallows model generalization
• Proposing joint MLE for central ranking, dispersion, and distance metric

Innovation

• Learns distance metric from data
• Uses L_alpha distances for flexibility
• Develops FPTAS for efficient sampling