$\ell_∞$-Bounds of the MLE in the BTL Model under General Comparison Graphs

Wanshan Li, Shamindra Shrotriya, Alessandro Rinaldo
Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence, PMLR 180:1178-1187, 2022.

Abstract

The Bradley-Terry-Luce (BTL) model is a popular statistical approach for estimating the global ranking of a collection of items using pairwise comparisons. To ensure accurate ranking, it is essential to obtain precise estimates of the model parameters in the $\ell_{\infty}$-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 Erd{ö}s-R{é}nyi comparison graphs, little is known about the performance of the maximum likelihood estimator (MLE) of the BTL model parameters in the $\ell_{\infty}$-loss under more general graph topologies. In this paper, we derive novel, general upper bounds on the $\ell_{\infty}$ estimation error of the BTL MLE that depend explicitly on the algebraic connectivity of the comparison graph, the maximal performance gap across items and the sample complexity. We demonstrate that the derived bounds perform well and in some cases are sharper compared to known results obtained using different loss functions and more restricted assumptions and graph topologies. We carefully compare our results to Yan et al. (2012), which is closest in spirit to our work. We further provide minimax lower bounds under $\ell_{\infty}$-error that nearly match the upper bounds over a class of sufficiently regular graph topologies. Finally, we study the implications of our $\ell_{\infty}$-bounds for efficient (offline) tournament design. We illustrate and discuss our findings through various examples and simulations.

Cite this Paper

BibTeX
@InProceedings{pmlr-v180-li22g, title = {$\ell_{∞}$-Bounds of the MLE in the BTL Model under General Comparison Graphs}, author = {Li, Wanshan and Shrotriya, Shamindra and Rinaldo, Alessandro}, booktitle = {Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence}, pages = {1178--1187}, year = {2022}, editor = {Cussens, James and Zhang, Kun}, volume = {180}, series = {Proceedings of Machine Learning Research}, month = {01--05 Aug}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v180/li22g/li22g.pdf}, url = {https://proceedings.mlr.press/v180/li22g.html}, abstract = {The Bradley-Terry-Luce (BTL) model is a popular statistical approach for estimating the global ranking of a collection of items using pairwise comparisons. To ensure accurate ranking, it is essential to obtain precise estimates of the model parameters in the $\ell_{\infty}$-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 Erd{ö}s-R{é}nyi comparison graphs, little is known about the performance of the maximum likelihood estimator (MLE) of the BTL model parameters in the $\ell_{\infty}$-loss under more general graph topologies. In this paper, we derive novel, general upper bounds on the $\ell_{\infty}$ estimation error of the BTL MLE that depend explicitly on the algebraic connectivity of the comparison graph, the maximal performance gap across items and the sample complexity. We demonstrate that the derived bounds perform well and in some cases are sharper compared to known results obtained using different loss functions and more restricted assumptions and graph topologies. We carefully compare our results to Yan et al. (2012), which is closest in spirit to our work. We further provide minimax lower bounds under $\ell_{\infty}$-error that nearly match the upper bounds over a class of sufficiently regular graph topologies. Finally, we study the implications of our $\ell_{\infty}$-bounds for efficient (offline) tournament design. We illustrate and discuss our findings through various examples and simulations.} }
Endnote
%0 Conference Paper %T $\ell_∞$-Bounds of the MLE in the BTL Model under General Comparison Graphs %A Wanshan Li %A Shamindra Shrotriya %A Alessandro Rinaldo %B Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence %C Proceedings of Machine Learning Research %D 2022 %E James Cussens %E Kun Zhang %F pmlr-v180-li22g %I PMLR %P 1178--1187 %U https://proceedings.mlr.press/v180/li22g.html %V 180 %X The Bradley-Terry-Luce (BTL) model is a popular statistical approach for estimating the global ranking of a collection of items using pairwise comparisons. To ensure accurate ranking, it is essential to obtain precise estimates of the model parameters in the $\ell_{\infty}$-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 Erd{ö}s-R{é}nyi comparison graphs, little is known about the performance of the maximum likelihood estimator (MLE) of the BTL model parameters in the $\ell_{\infty}$-loss under more general graph topologies. In this paper, we derive novel, general upper bounds on the $\ell_{\infty}$ estimation error of the BTL MLE that depend explicitly on the algebraic connectivity of the comparison graph, the maximal performance gap across items and the sample complexity. We demonstrate that the derived bounds perform well and in some cases are sharper compared to known results obtained using different loss functions and more restricted assumptions and graph topologies. We carefully compare our results to Yan et al. (2012), which is closest in spirit to our work. We further provide minimax lower bounds under $\ell_{\infty}$-error that nearly match the upper bounds over a class of sufficiently regular graph topologies. Finally, we study the implications of our $\ell_{\infty}$-bounds for efficient (offline) tournament design. We illustrate and discuss our findings through various examples and simulations.
APA
Li, W., Shrotriya, S. & Rinaldo, A.. (2022). $\ell_∞$-Bounds of the MLE in the BTL Model under General Comparison Graphs. Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence, in Proceedings of Machine Learning Research 180:1178-1187 Available from https://proceedings.mlr.press/v180/li22g.html.

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