June 2022 Partial recovery for top-k ranking: Optimality of MLE and SubOptimality of the spectral method
Pinhan Chen, Chao Gao, Anderson Y. Zhang
Author Affiliations +
Ann. Statist. 50(3): 1618-1652 (June 2022). DOI: 10.1214/21-AOS2166


Given partially observed pairwise comparison data generated by the Bradley–Terry–Luce (BTL) model, we study the problem of top-k ranking. That is, to optimally identify the set of top-k players. We derive the minimax rate with respect to a normalized Hamming loss. This provides the first result in the literature that characterizes the partial recovery error in terms of the proportion of mistakes for top-k ranking. We also derive the optimal signal to noise ratio condition for the exact recovery of the top-k set. The maximum likelihood estimator (MLE) is shown to achieve both optimal partial recovery and optimal exact recovery. On the other hand, we show another popular algorithm, the spectral method, is in general suboptimal. Our results complement the recent work (Ann. Statist. 47 (2019) 2204–2235) that shows both the MLE and the spectral method achieve the optimal sample complexity for exact recovery. It turns out the leading constants of the sample complexity are different for the two algorithms. Another contribution that may be of independent interest is the analysis of the MLE without any penalty or regularization for the BTL model. This closes an important gap between theory and practice in the literature of ranking.

Funding Statement

The first and second authors were supported in part by NSF CAREER award DMS-1847590 and NSF Grant CCF-1934931.
The third author was supported in part by NSF Grant DMS-2112988.


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Pinhan Chen. Chao Gao. Anderson Y. Zhang. "Partial recovery for top-k ranking: Optimality of MLE and SubOptimality of the spectral method." Ann. Statist. 50 (3) 1618 - 1652, June 2022. https://doi.org/10.1214/21-AOS2166


Received: 1 July 2021; Revised: 1 December 2021; Published: June 2022
First available in Project Euclid: 16 June 2022

MathSciNet: MR4441134
zbMATH: 07547944
Digital Object Identifier: 10.1214/21-AOS2166

Primary: 62F07

Keywords: exact recovery , MLE , pairwise comparison , partial recovery , Spectral method , Top-k ranking

Rights: Copyright © 2022 Institute of Mathematical Statistics


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Vol.50 • No. 3 • June 2022
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