Optimal Full Ranking from Pairwise Comparisons
This work addresses the fundamental statistical challenge of optimal ranking from noisy comparisons, which is incremental in deriving minimax rates for a specific distance metric.
The paper tackles the problem of ranking players from pairwise comparisons under the Bradley-Terry-Luce model, deriving the minimax rate with respect to Kendall's tau distance and showing a transition between exponential and polynomial rates based on signal-to-noise ratio. It proposes a divide-and-conquer algorithm that achieves this minimax rate by grouping players and computing local maximum likelihood estimates.
We consider the problem of ranking $n$ players from partial pairwise comparison data under the Bradley-Terry-Luce model. For the first time in the literature, the minimax rate of this ranking problem is derived with respect to the Kendall's tau distance that measures the difference between two rank vectors by counting the number of inversions. The minimax rate of ranking exhibits a transition between an exponential rate and a polynomial rate depending on the magnitude of the signal-to-noise ratio of the problem. To the best of our knowledge, this phenomenon is unique to full ranking and has not been seen in any other statistical estimation problem. To achieve the minimax rate, we propose a divide-and-conquer ranking algorithm that first divides the $n$ players into groups of similar skills and then computes local MLE within each group. The optimality of the proposed algorithm is established by a careful approximate independence argument between the two steps.