Entrywise Error Bounds for Spectral Ranking with Semi-Random Adversaries
2026-05-22 • Machine Learning
Machine Learning
AI summaryⓘ
The authors study a method to rank items by looking at pairwise comparisons, focusing on a common model called Bradley-Terry-Luce (BTL). They show that when the comparisons come from a network that is changed by an adversary to favor some pairs more, the usual spectral ranking method can perform poorly. However, by adjusting the weights of the edges in the network, the authors find that the ranking accuracy can be improved to match cases where comparisons are evenly sampled. Their numerical tests back up these theoretical results.
Bradley-Terry-Luce modelpairwise comparisonsspectral algorithmmaximum likelihood estimationrandom graphssemi-random adversaryspectral gapranking algorithms
Authors
Dongmin Lee, Anuran Makur, Japneet Singh
Abstract
Bradley-Terry-Luce (BTL) model estimation is a well-established strategy to rank a collection of items given a dataset of pairwise comparisons. Although the theoretical performance of BTL estimation methods, such as spectral and maximum likelihood estimation, is well studied in the regime of uniformly sampled graphs, generalizing such results to a wider class of random graphs has proved challenging. In this work, we investigate the entry-wise error of spectral algorithms against a semi-random adversary that can arbitrarily boost the sampling probabilities of certain edges. We find that the performance of the unweighted spectral method is heavily dependent on the spectral properties of the generated graph. Furthermore, we show that asymptotic performance approaching that of uniformly sampled graphs can be recovered by appropriately reweighting the observed edges to counteract the adversary and restore the spectral gap. Finally, we provide numerical simulations that support our theoretical findings.