Problem
Research questions and friction points this paper is trying to address.
Bradley-Terry model
maximum likelihood estimator
comparison graphs
uniform consistency
asymptotic normality
Bradley-Terry model under general comparison graphs
🤖 AI Summary
This study addresses the reliance of existing Bradley-Terry model theory on graph homogeneity assumptions, which limits its applicability to practical scenarios with a growing number of items. By leveraging effective resistance and spectral analysis of the graph Laplacian, this work establishes the consistency of the maximum likelihood estimator (MLE) under general deterministic comparison designs and derives tight error bounds. Furthermore, it proves the asymptotic normality of utility difference estimators under specific conditions. This research overcomes traditional homogeneity constraints by developing a unified consistency theory applicable to arbitrary graph structures. Additionally, it provides explicit connectivity thresholds for independent-edge random graph designs, thereby substantially expanding the applicability boundaries of preference learning models.
📝 Abstract
The Bradley-Terry model is a parametric model for ranking from pairwise comparisons. Existing asymptotic theory for the maximum likelihood estimator (MLE) in regimes where the number of objects grows often requires homogeneity assumptions or compatibility conditions on comparison graphs, which limits its applicability to many practical settings. In this work, we establish uniform consistency of the MLE under general deterministic comparison designs. Our pairwise error bound consists of a pair-specific term governed by the effective resistance between the corresponding objects and a global term governed by the spectral gap of the unnormalized graph Laplacian. The bound is tight up to logarithmic factors for certain graph sequences. For any prespecified sequence of pairs satisfying an additional balancing condition, we further establish asymptotic normality of the corresponding estimated utility differences. As an application, we obtain uniform consistency for independent-edge random graph designs whenever the minimum edge probability exceeds the Erd\H{o}s-R\'enyi connectivity threshold by a logarithmic factor, as well as asymptotic normality under additional balancing conditions.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Bradley-Terry model
maximum likelihood estimator
effective resistance
spectral gap
asymptotic normality
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