• Corpus ID: 239050145

# The Performance of the MLE in the Bradley-Terry-Luce Model in $\ell_{\infty}$-Loss and under General Graph Topologies

@inproceedings{Li2021ThePO,
title={The Performance of the MLE in the Bradley-Terry-Luce Model in \$\ell\_\{\infty\}\$-Loss and under General Graph Topologies},
author={Wanshan Li and Shamindra Shrotriya and Alessandro Rinaldo},
year={2021}
}
• Published 20 October 2021
• Computer Science, Mathematics
The Bradley-Terry-Luce (BTL) model is a popular statistical approach for estimating the global ranking of a collection of items of interest using pairwise comparisons. To ensure accurate ranking, it is essential to obtain precise estimates of the model parameters in the `∞-loss. The difficulty of this task depends crucially on the topology of the pairwise comparison graph over the given items. However, beyond very few well-studied cases, such as the complete and Erdös-Rényi comparison graphs…

## References

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• Computer Science
J. Mach. Learn. Res.
• 2016
This work considers parametric ordinal models for pairwise comparison data involving a latent vector w* e Rd that represents the "qualities" of the d items being compared; this class of models includes the two most widely used parametric models|the Bradley-Terry-Luce (BTL) and the Thurstone models.
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This work derives the minimax rate, the first result in the literature that characterizes the partial recovery error in terms of the proportion of mistakes for top-$k$ ranking, and derives the optimal signal to noise ratio condition for the exact recovery of the top-k$set. Asymptotics when the number of parameters tends to infinity in the Bradley-Terry model for paired comparisons • Mathematics • 1999 We are concerned here with establishing the consistency and asymptotic normality for the maximum likelihood estimator of a merit vector (u 0 , ..., u t ), representing the merits of t+1 teams Graph Resistance and Learning from Pairwise Comparisons • Computer Science ICML • 2019 It is proved that, after a known transition period, the relevant graph-theoretic quantity is the square root of the resistance of the comparison graph, and it is shown that the performance guarantee of the algorithm, both in terms of the graph and the skewness of the item quality distribution, outperforms earlier results. Spectral Method and Regularized MLE Are Both Optimal for Top-$K$Ranking • Computer Science Annals of statistics • 2019 It is demonstrated that under a natural random sampling model, the spectral method alone, or the regularized MLE alone, is minimax optimal in terms of the sample complexity - the number of paired comparisons needed to ensure exact top-K identification, for the fixed dynamic range regime. Minimax-optimal Inference from Partial Rankings • Computer Science, Mathematics NIPS • 2014 It is shown that even if one applies the mismatched maximum likelihood estimator that assumes independence (on pairwise comparisons that are now dependent due to rank-breaking), minimax optimal performance is still achieved up to a logarithmic factor. Minimax Rate for Learning From Pairwise Comparisons in the BTL Model • Computer Science ICML • 2020 It is shown that the determination of the minimax rate is achieved by a simple algorithm based on weighted least squares, with weights determined from the empirical outcomes of the comparisons, which can be implemented in nearly linear time in the total number of comparisons. Accelerated Spectral Ranking • Computer Science ICML • 2018 This paper designs a provably faster spectral ranking algorithm, which it is called accelerated spectral ranking (ASR), that is also consistent under the MNL/BTL models, and gives the first general sample complexity bounds for recovering the parameters of theMNL model from multiway comparisons under any (connected) comparison graph. Spectral MLE: Top-K Rank Aggregation from Pairwise Comparisons • Computer Science ICML • 2015 This paper describes the minimax limits on identifiability of top-$K$ranked items, in the presence of random and non-adaptive sampling, and proposes a nearly linear-time ranking scheme, called Spectral MLE, that allows perfect top-K$ item identification under minimal sample complexity.
MM algorithms for generalized Bradley-Terry models
The Bradley-Terry model for paired comparisons is a simple and muchstudied means to describe the probabilities of the possible outcomes when individuals are judged against one another in pairs. Among