When can we rank well from comparisons of \(O(n\log(n))\) non-actively chosen pairs?

Ranking from pairwise comparisons is a ubiquitous problem and has been studied in disciplines ranging from statistics to operations research and from theoretical computer science to machine learning. Here we consider a general setting where outcomes of pairwise comparisons between items i and j are drawn probabilistically by flipping a coin with unknown bias Pij , and ask under what conditions on these unknown probabilities one can learn a good ranking from comparisons of only O(n log n) non-actively chosen pairs. Recent work has established this is possible under the Bradley-Terry-Luce (BTL) and noisy permutation (NP) models. Here we introduce a broad family of ‘low-rank’ conditions on the probabilities Pij under which the resulting preference matrix P has low rank under some link function, and show these conditions encompass the BTL and Thurstone classes as special cases, but are considerably more general. We then give a new algorithm called low-rank pairwise ranking (LRPR) which provably learns a good ranking from comparisons of only O(n log n) randomly chosen comparisons under such low-rank models. Our algorithm and analysis make use of tools from the theory of low-rank matrix completion, and provide a new perspective on the problem of ranking from pairwise comparisons in non-active settings.