Spectral MLE: Top-$K$ Rank Aggregation from Pairwise Comparisons

TL;DR

This paper introduces Spectral MLE, a nearly linear-time algorithm for top-$K$ rank aggregation from pairwise comparisons under the BTL model, achieving minimax optimal sample complexity and practical effectiveness.

Contribution

It proposes a novel spectral MLE method that refines initial estimates with coordinate-wise MLEs for accurate top-$K$ ranking, matching minimax limits.

Findings

Spectral MLE achieves near-minimal sample complexity for top-$K$ identification.

The method is computationally efficient, running in nearly linear time.

Numerical experiments validate the practical effectiveness of Spectral MLE.

Abstract

This paper explores the preference-based top-$K$ rank aggregation problem. Suppose that a collection of items is repeatedly compared in pairs, and one wishes to recover a consistent ordering that emphasizes the top-$K$ ranked items, based on partially revealed preferences. We focus on the Bradley-Terry-Luce (BTL) model that postulates a set of latent preference scores underlying all items, where the odds of paired comparisons depend only on the relative scores of the items involved. We characterize the minimax limits on identifiability of top-$K$ ranked items, in the presence of random and non-adaptive sampling. Our results highlight a separation measure that quantifies the gap of preference scores between the $K^{\text{th}}$ and $(K+1)^{\text{th}}$ ranked items. The minimum sample complexity required for reliable top-$K$ ranking scales inversely with the separation measure irrespective of other preference distribution metrics. To approach this minimax limit, we propose a nearly linear-time ranking scheme, called \emph{Spectral MLE}, that returns the indices of the top-$K$ items in accordance to a careful score estimate. In a nutshell, Spectral MLE starts with an initial score estimate with minimal squared loss (obtained via a spectral method), and then successively refines each component with the assistance of coordinate-wise MLEs. Encouragingly, Spectral MLE allows perfect top-$K$ item identification under minimal sample complexity. The practical applicability of Spectral MLE is further corroborated by numerical experiments.