Title: Spectral Method and Regularized MLE Are Both Optimal for Top-$K$ Ranking,
Abstract: This paper is concerned with the problem of top-$K$ ranking from pairwise
comparisons. Given a collection of $n$ items and a few pairwise comparisons
across them, one wishes to identify the set of $K$ items that receive the
highest ranks. To tackle this problem, we adopt the logistic parametric model
--- the Bradley-Terry-Luce model, where each item is assigned a latent
preference score, and where the outcome of each pairwise comparison depends
solely on the relative scores of the two items involved. Recent works have made
significant progress towards characterizing the performance (e.g. the mean
square error for estimating the scores) of several classical methods, including
the spectral method and the maximum likelihood estimator (MLE). However, where
they stand regarding top-$K$ ranking remains unsettled.
We demonstrate that under a natural random sampling model, the spectral
method alone, or the regularized MLE alone, is minimax optimal in terms of the
sample complexity --- the number of paired comparisons needed to ensure exact
top-$K$ identification, for the fixed dynamic range regime. This is
accomplished via optimal control of the entrywise error of the score estimates.
We complement our theoretical studies by numerical experiments, confirming that
both methods yield low entrywise errors for estimating the underlying scores.
Our theory is established via a novel leave-one-out trick, which proves
effective for analyzing both iterative and non-iterative procedures. Along the
way, we derive an elementary eigenvector perturbation bound for probability
transition matrices, which parallels the Davis-Kahan $\sin\Theta$ theorem for
symmetric matrices. This also allows us to close the gap between the $\ell_2$
error upper bound for the spectral method and the minimax lower limit.