The asymptotics of ranking algorithms

TL;DR

This paper investigates the theoretical challenges of supervised ranking with partial preferences, demonstrating inconsistencies in common surrogate losses, and proposes a new aggregation-based approach with proven asymptotic consistency.

Contribution

It introduces a novel aggregation-based ranking method using U-statistics and provides the first asymptotic analysis showing its consistency in the partial preference setting.

Findings

Common surrogate losses are inconsistent for partial preferences.

The new aggregation method achieves asymptotic consistency.

Experimental results support the theoretical advantages of the proposed approach.

Abstract

We consider the predictive problem of supervised ranking, where the task is to rank sets of candidate items returned in response to queries. Although there exist statistical procedures that come with guarantees of consistency in this setting, these procedures require that individuals provide a complete ranking of all items, which is rarely feasible in practice. Instead, individuals routinely provide partial preference information, such as pairwise comparisons of items, and more practical approaches to ranking have aimed at modeling this partial preference data directly. As we show, however, such an approach raises serious theoretical challenges. Indeed, we demonstrate that many commonly used surrogate losses for pairwise comparison data do not yield consistency; surprisingly, we show inconsistency even in low-noise settings. With these negative results as motivation, we present a new approach to supervised ranking based on aggregation of partial preferences, and we develop $U$-statistic-based empirical risk minimization procedures. We present an asymptotic analysis of these new procedures, showing that they yield consistency results that parallel those available for classification. We complement our theoretical results with an experiment studying the new procedures in a large-scale web-ranking task.