$\ell_{\infty}$-Bounds of the MLE in the BTL Model under General Comparison Graphs

TL;DR

This paper establishes new bounds on the maximum likelihood estimator's accuracy in the Bradley-Terry-Luce model across general comparison graphs, linking error to graph connectivity and sample size.

Contribution

It provides the first general upper bounds on the $\, ext{l}_ ext{infinity} ext{-error of the MLE in the BTL model for arbitrary comparison graphs, including minimax lower bounds.

Findings

Bounds depend on graph algebraic connectivity and sample complexity.

Bounds are sharper than previous results in some cases.

Implications for tournament design and ranking accuracy.

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\"os-R\'enyi 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.