# Efficient Ranking from Pairwise Comparisons

;
JMLR W&CP 28
(3)
:
109–117, 2013

## Abstract

The ranking of \(n\) objects based on pairwise comparisons is a core machine learning problem, arising in recommender systems, ad placement, player ranking, biological applications and others. In many practical situations the true pairwise comparisons cannot be actively measured, but a subset of all \(n(n-1)/2\) comparisons is passively and noisily observed. Optimization algorithms (e.g., the SVM) could be used to predict a ranking with fixed expected Kendall tau distance, while achieving an \(\Omega(n)\) lower bound on the corresponding sample complexity. However, due to their centralized structure they are difficult to extend to online or distributed settings. In this paper we show that much simpler algorithms can match the same \(\Omega(n)\) lower bound in expectation. Furthermore, if an average of \(O(n\log(n))\) binary comparisons are measured, then one algorithm recovers the true ranking in a uniform sense, while the other predicts the ranking more accurately near the top than the bottom. We discuss extensions to online and distributed ranking, with benefits over traditional alternatives.