  In this paper, we study a popular method for inference of the Bradley-Terry
model parameters, namely the MM algorithm, for maximum likelihood estimation
and maximum a posteriori probability estimation. This class of models includes
the Bradley-Terry model of paired comparisons, the Rao-Kupper model of paired
comparisons allowing for tie outcomes, the Luce choice model, and the
Plackett-Luce ranking model. We establish tight characterizations of the
convergence rate for the MM algorithm, and show that it is essentially
equivalent to that of a gradient descent algorithm. For the maximum likelihood
estimation, the convergence is shown to be linear with the rate crucially
determined by the algebraic connectivity of the matrix of item pair
co-occurrences in observed comparison data. For the Bayesian inference, the
convergence rate is also shown to be linear, with the rate determined by a
parameter of the prior distribution in a way that can make the convergence
arbitrarily slow for small values of this parameter. We propose a simple
modification of the classical MM algorithm that avoids the observed slow
convergence issue and accelerates the convergence. The key component of the
accelerated MM algorithm is a parameter rescaling performed at each iteration
step that is carefully chosen based on theoretical analysis and
characterisation 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 proposed
method.
