The Lovasz-Bregman Divergence and connections to rank aggregation, clustering, and web ranking

TL;DR

This paper extends Lovasz-Bregman divergences to better measure distortions between scores and orderings, connecting them to rank aggregation, clustering, and web ranking, and demonstrating their ability to unify and enhance existing ranking metrics.

Contribution

It introduces new properties of Lovasz-Bregman divergences, linking them to permutation metrics and ranking measures like NDCG and AUC, and shows their application in web ranking and learning to rank.

Findings

LB divergences relate to permutation metrics like Kendall-$\tau$

They unify and extend common ranking measures such as NDCG and AUC

They naturally incorporate confidence in orderings for ranking applications

Abstract

We extend the recently introduced theory of Lovasz-Bregman (LB) divergences (Iyer & Bilmes, 2012) in several ways. We show that they represent a distortion between a 'score' and an 'ordering', thus providing a new view of rank aggregation and order based clustering with interesting connections to web ranking. We show how the LB divergences have a number of properties akin to many permutation based metrics, and in fact have as special cases forms very similar to the Kendall-$\tau$ metric. We also show how the LB divergences subsume a number of commonly used ranking measures in information retrieval, like the NDCG and AUC. Unlike the traditional permutation based metrics, however, the LB divergence naturally captures a notion of "confidence" in the orderings, thus providing a new representation to applications involving aggregating scores as opposed to just orderings. We show how a number of recently used web ranking models are forms of Lovasz-Bregman rank aggregation and also observe that a natural form of Mallow's model using the LB divergence has been used as conditional ranking models for the 'Learning to Rank' problem.