Open Access
August 2019 Spectral method and regularized MLE are both optimal for top-$K$ ranking
Yuxin Chen, Jianqing Fan, Cong Ma, Kaizheng Wang
Ann. Statist. 47(4): 2204-2235 (August 2019). DOI: 10.1214/18-AOS1745


This paper is concerned with the problem of top-$K$ ranking from pairwise comparisons. Given a collection of $n$ items and a few pairwise comparisons across them, one wishes to identify the set of $K$ items that receive the highest ranks. To tackle this problem, we adopt the logistic parametric model—the Bradley–Terry–Luce model, where each item is assigned a latent preference score, and where the outcome of each pairwise comparison depends solely on the relative scores of the two items involved. Recent works have made significant progress toward characterizing the performance (e.g., the mean square error for estimating the scores) of several classical methods, including the spectral method and the maximum likelihood estimator (MLE). However, where they stand regarding top-$K$ ranking remains unsettled.

We demonstrate that under a natural random sampling model, the spectral method alone, or the regularized MLE alone, is minimax optimal in terms of the sample complexity—the number of paired comparisons needed to ensure exact top-$K$ identification, for the fixed dynamic range regime. This is accomplished via optimal control of the entrywise error of the score estimates. We complement our theoretical studies by numerical experiments, confirming that both methods yield low entrywise errors for estimating the underlying scores. Our theory is established via a novel leave-one-out trick, which proves effective for analyzing both iterative and noniterative procedures. Along the way, we derive an elementary eigenvector perturbation bound for probability transition matrices, which parallels the Davis–Kahan $\mathop{\mathrm{sin}}\nolimits \Theta $ theorem for symmetric matrices. This also allows us to close the gap between the $\ell_{2}$ error upper bound for the spectral method and the minimax lower limit.


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Yuxin Chen. Jianqing Fan. Cong Ma. Kaizheng Wang. "Spectral method and regularized MLE are both optimal for top-$K$ ranking." Ann. Statist. 47 (4) 2204 - 2235, August 2019.


Received: 1 August 2017; Revised: 1 July 2018; Published: August 2019
First available in Project Euclid: 21 May 2019

zbMATH: 07082284
MathSciNet: MR3953449
Digital Object Identifier: 10.1214/18-AOS1745

Primary: 62F07
Secondary: 62B10

Keywords: entrywise perturbation , leave-one-out analysis , pairwise comparisons , regularized MLE , reversible Markov chains , Spectral method , Top-$K$ ranking

Rights: Copyright © 2019 Institute of Mathematical Statistics

Vol.47 • No. 4 • August 2019
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