Rank Centrality: Ranking from Pairwise Comparisons

Abstract

The question of aggregating pair-wise comparisons to obtain a global ranking over a collection of objects has been of interest for a very long time: be it ranking of online gamers (e.g. MSR's TrueSkill system) and chess players, aggregating social opinions, or deciding which product to sell based on transactions. In most settings, in addition to obtaining a ranking, finding 'scores' for each object (e.g. player's rating) is of interest for understanding the intensity of the preferences.In this paper, we propose Rank Centrality, an iterative rank aggregation algorithm † for discovering scores for objects (or items) from pair-wise comparisons. The algorithm has a natural random walk interpretation over the graph of objects with an edge present between a pair of objects if they are compared; the score, which we call Rank Centrality, of an object turns out to be its stationary probability under this random walk.To study the efficacy of the algorithm, we consider the popular Bradley-Terry-Luce (BTL) model (equivalent to the Multinomial Logit (MNL) for pair-wise comparisons) in which each object has an associated score which determines the probabilistic outcomes of pair-wise comparisons between objects. In terms of the pair-wise marginal probabilities, which is the main subject of this paper, the MNL model and the BTL model are identical. We bound the finite sample error rates between the scores assumed by the BTL model and those estimated by our algorithm. In particular, the number of samples required to learn the score well with high probability depends on the structure of the comparison graph. When the Laplacian of the comparison graph has a strictly positive spectral gap, e.g. each item is compared to a subset of randomly chosen items, this leads to dependence on the number of samples that is nearly order-optimal.Experimental evaluations on synthetic datasets generated according to the BTL model show that our algorithm performs as well as the Maximum Likelihood estimator for that model and outperforms other popular ranking algorithms.

Peer-Assisted Individual Assessment in a multi-agent system

“…Recent years are seeing renewed interest in ranking by pairwise comparisons (see, e.g. Heckel et al, 2016;Jamieson and Nowak, 2011;Negahban et al, 2012;Wauthier et al, 2013, and references therein). In these works a central authority observes the whole collection of outcomes, and usually directs the measurement process.…”

A Nearly Instance Optimal Algorithm for Top-k Ranking under the Multinomial Logit Model

“…When the size of S t is 2 (i.e., l = 2), the MNL model reduces to Bradley-Terry model [3], which has been widely studied in rank aggregation literature in machine learning (see, e.g., [22,16,23,10]). In fact, the MNL model has a simple probabilistic interpretation as follows [28].…”

“…Most existing works focus on the non-active setting: the pairs of items for comparisons are fixed (or chosen completely at random) and the algorithm cannot adaptively choose the next pair for querying. In this non-active ranking setup, when the goal is to obtain a global ranking over all the items, Negahban et al [22] proposed the RankCentrality algorithm under the popular Bradley-Terry model, which is a special case of the MNL model for pairwise comparisons. Lu and Boutilier [18] proposed a ranking algorithm under the Mallows model.…”

A practical divide-and-conquer approach for preference-based learning to rank

“…We compare FUZZY-SORT with several other counterparts being FAS-PIVOT, averaged-MERGE-SORT 3 , a Markov chained based approach (RANK-CENTRALITY) by Negahban et al [22], and the sort-by-degree heuristic in raw. Note the latter two algorithms query all quadratic pairs of preference relations before execution, while the previous three have performance that scale with the number of queries they are allotted to make.…”

“…Despite for some works that are based on heuristics [22], existing works on solving the weighted Min-FAST problem often focus on the theoretical perspective. For instance, [18] proves that a simple approach [17], which is of time complexity quadratic to the number of test instances, is a 2-approximation of the optimal solution for a special ranking task called bipartite ranking.…”