Open Access
April 2020 Worst-case versus average-case design for estimation from partial pairwise comparisons
Ashwin Pananjady, Cheng Mao, Vidya Muthukumar, Martin J. Wainwright, Thomas A. Courtade
Ann. Statist. 48(2): 1072-1097 (April 2020). DOI: 10.1214/19-AOS1838


Pairwise comparison data arises in many domains, including tournament rankings, web search and preference elicitation. Given noisy comparisons of a fixed subset of pairs of items, we study the problem of estimating the underlying comparison probabilities under the assumption of strong stochastic transitivity (SST). We also consider the noisy sorting subclass of the SST model. We show that when the assignment of items to the topology is arbitrary, these permutation-based models, unlike their parametric counterparts, do not admit consistent estimation for most comparison topologies used in practice. We then demonstrate that consistent estimation is possible when the assignment of items to the topology is randomized, thus establishing a dichotomy between worst-case and average-case designs. We propose two computationally efficient estimators in the average-case setting and analyze their risk, showing that it depends on the comparison topology only through the degree sequence of the topology. We also provide explicit classes of graphs for which the rates achieved by these estimators are optimal. Our results are corroborated by simulations on multiple comparison topologies.


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Ashwin Pananjady. Cheng Mao. Vidya Muthukumar. Martin J. Wainwright. Thomas A. Courtade. "Worst-case versus average-case design for estimation from partial pairwise comparisons." Ann. Statist. 48 (2) 1072 - 1097, April 2020.


Received: 1 October 2017; Revised: 1 October 2018; Published: April 2020
First available in Project Euclid: 26 May 2020

zbMATH: 07241581
MathSciNet: MR4102688
Digital Object Identifier: 10.1214/19-AOS1838

Primary: 62F07 , 62J15

Keywords: pairwise comparisons , strong stochastic transitivity , structured matrix completion

Rights: Copyright © 2020 Institute of Mathematical Statistics

Vol.48 • No. 2 • April 2020
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