Annals of Statistics

Spectral method and regularized MLE are both optimal for top-$K$ ranking

Yuxin Chen, Jianqing Fan, Cong Ma, and Kaizheng Wang

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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.

Article information

Ann. Statist., Volume 47, Number 4 (2019), 2204-2235.

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F07: Ranking and selection
Secondary: 62B10: Information-theoretic topics [See also 94A17]

Top-$K$ ranking pairwise comparisons spectral method regularized MLE entrywise perturbation leave-one-out analysis reversible Markov chains


Chen, Yuxin; Fan, Jianqing; Ma, Cong; Wang, Kaizheng. Spectral method and regularized MLE are both optimal for top-$K$ ranking. Ann. Statist. 47 (2019), no. 4, 2204--2235. doi:10.1214/18-AOS1745.

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Supplemental materials

  • Additional Proofs. Additional proofs of the results in the paper can be found in the Supplementary Material.