Clustering and Inference From Pairwise Comparisons

Abstract

Given a set of pairwise comparisons, the classical ranking problem computes a single ranking that best represents the preferences of all users. In this paper, we study the problem of inferring individual preferences, arising in the context of making personalized recommendations. In particular, we assume that there are n users of r types; users of the same type provide similar pairwise comparisons for m items according to the BradleyTerry model. We propose an efficient algorithm that accurately estimates the individual preferences for almost all users, if there are r max{m, n} log m log 2 n pairwise comparisons per type, which is near optimal in sample complexity when r only grows logarithmically with m or n. Our algorithm has three steps: first, for each user, compute the net-win vector which is a projection of its m 2 -dimensional vector of pairwise comparisons onto an m-dimensional linear subspace; second, cluster the users based on the net-win vectors; third, estimate a single preference for each cluster separately. The net-win vectors are much less noisy than the high dimensional vectors of pairwise comparisons and clustering is more accurate after the projection as confirmed by numerical experiments. Moreover, we show that, when a cluster is only approximately correct, the maximum likelihood estimation for the Bradley-Terry model is still close to the true preference.

Clustering and Inference From Pairwise Comparisons

“…For the complete algorithm, see [5]. The basic idea is to estimate θ in two steps: cluster the users and then estimate a score vector for each cluster separately.…”

“…We overcome this difficulty by representing each user u by its net-win vector Su, defined in the previous section, instead. The effectiveness of Su lies in the nontrivial fact that E [Ru] are close to an (m − 1)-dimensional linear subspace [5], therefore one can reduce the noise of Ru by projecting it onto this linear subspace.…”

“…First, for each user u, the comparison vector Ru lies in a high dimensional space , but only a small number of entries are observed. If one directly clusters the vectors Ru, the resulted clusters will be very noisy, which is demonstrated in our numerical experiment [5]. Second, existing algorithms for estimating the score vector [3,4] assume the users to have the same preference, and it is unclear how to estimate the score vectors when the clusters are only approximately recovered.…”

“…Let Du,ij = I R (2) u,ij =1 and Du,ji = I R (2) u,ij =−1 for any u and i < j. For users in cluster C k , the estimated score vector is given by θ k = arg maxγ L k (γ), where 5] for any arbitrarily small constant b0 > 0. Then there exists a constant C > 0 such that…”

Model-Based Learning from Preference Data

“…The BT model has been generalized in several directions (see e.g. Davidson, 1970;Agresti, 1996;Wu et al, 2015). Maximum likelihood estimation is typically performed through iterative algorithms and Majorized -Minimization algorithms (Hunter, 2004).…”