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In this paper, we study a popular method for inference of the Bradley-Terrymodel parameters, namely the MM algorithm, for maximum likelihood estimationand maximum a posteriori probability estimation. This class of models includesthe Bradley-Terry model of paired comparisons, the Rao-Kupper model of pairedcomparisons allowing for tie outcomes, the Luce choice model, and thePlackett-Luce ranking model. We establish tight characterizations of theconvergence rate for the MM algorithm, and show that it is essentiallyequivalent to that of a gradient descent algorithm. For the maximum likelihoodestimation, the convergence is shown to be linear with the rate cruciallydetermined by the algebraic connectivity of the matrix of item pairco-occurrences in observed comparison data. For the Bayesian inference, theconvergence rate is also shown to be linear, with the rate determined by aparameter of the prior distribution in a way that can make the convergencearbitrarily slow for small values of this parameter. We propose a simplemodification of the classical MM algorithm that avoids the observed slowconvergence issue and accelerates the convergence. The key component of theaccelerated MM algorithm is a parameter rescaling performed at each iterationstep that is carefully chosen based on theoretical analysis andcharacterisation of the convergence rate.  Our experimental results, performed on both synthetic and real-world data,demonstrate the identified slow convergence issue of the classic MM algorithm,and show that significant efficiency gains can be obtained by our new proposedmethod.

Accelerated MM Algorithms for Ranking Scores Inference from Comparison Data