# A Statistical Convergence Perspective of Algorithms for Rank Aggregation from Pairwise Data

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Arun Rajkumar,
Shivani Agarwal
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Proceedings of the 31st International Conference on Machine Learning, PMLR 32(1):118-126, 2014.

#### Abstract

There has been much interest recently in the problem of rank aggregation from pairwise data. A natural question that arises is: under what sorts of statistical assumptions do various rank aggregation algorithms converge to an ‘optimal’ ranking? In this paper, we consider this question in a natural setting where pairwise comparisons are drawn randomly and independently from some underlying probability distribution. We first show that, under a ‘time-reversibility’ or Bradley-Terry-Luce (BTL) condition on the distribution generating the outcomes of the pairwise comparisons, the rank centrality (PageRank) and least squares (HodgeRank) algorithms both converge to an optimal ranking. Next, we show that a matrix version of the Borda count algorithm, and more surprisingly, an algorithm which performs maximal likelihood estimation under a BTL assumption, both converge to an optimal ranking under a ‘low-noise’ condition that is strictly more general than BTL. Finally, we propose a new SVM-based algorithm for rank aggregation from pairwise data, and show that this converges to an optimal ranking under an even more general condition that we term ‘generalized low-noise’. In all cases, we provide explicit sample complexity bounds for exact recovery of an optimal ranking. Our experiments confirm our theoretical findings and help to shed light on the statistical behavior of various rank aggregation algorithms.

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