Rank aggregation based on pairwise comparisons over a set of items has a wide range of applications. Although considerable research has been devoted to the development of rank aggregation algorithms, one basic question is how to efficiently collect a large amount of high-quality pairwise comparisons for the ranking purpose. Because of the advent of many crowdsourcing services, a crowd of workers are often hired to conduct pairwise comparisons with a small monetary reward for each pair they compare. Since different workers have different levels of reliability and different pairs have different levels of ambiguity, it is desirable to wisely allocate the limited budget for comparisons among the pairs of items and workers so that the global ranking can be accurately inferred from the comparison results. To this end, we model the active sampling problem in crowdsourced ranking as a Bayesian Markov decision process, which dynamically selects item pairs and workers to improve the ranking accuracy under a budget constraint. We further develop a computationally efficient sampling policy based on knowledge gradient as well as a moment matching technique for posterior approximation. Experimental evaluations on both synthetic and real data show that the proposed policy achieves high ranking accuracy with a lower labeling cost.
The dataset of partial rankings over items can be generated from a variety of sources including crowdsourcing services , online competition games (e.g., Microsoft's TrueSkill system ), and online users' activities such as browsing, clicking and transactions that reveal certain preferences. Learning a global ranking of a large set of items by aggregating a collection of partial rankings preferences has been an active research area for the past ten years (see, e.g., @cite_10 @cite_2 @cite_6 @cite_0 @cite_8 @cite_18 @cite_17 @cite_5 ). However, most work on rank aggregation considers a static estimation problem --- inferring a global ranking based on a pre-existing dataset. The problem we consider here is related to but significantly different from these works because we model crowdsourced ranking as a dynamic procedure where the inference of ranking and collection of data proceed concurrently and influence each other.
@cite_18 @cite_8 @cite_6 @cite_0 @cite_2 @cite_5 @cite_10 @cite_17
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There has been much interest recently in the problem of rank aggregation from pairwise data. A natural question that arises is: under what sorts of statistical assumptions do various rank aggregation algorithms converge to an 'optimal' ranking? In this paper, we consider this question in a natural setting where pairwise comparisons are drawn randomly and independently from some underlying probability distribution. We first show that, under a 'time-reversibility' or Bradley-Terry-Luce (BTL) condition on the distribution, the rank centrality (PageRank) and least squares (HodgeRank) algorithms both converge to an optimal ranking. Next, we show that a matrix version of the Borda count algorithm, and more surprisingly, an algorithm which performs maximum likelihood estimation under a BTL assumption, both converge to an optimal ranking under a 'low-noise' condition that is strictly more general than BTL. Finally, we propose a new SVM-based algorithm for rank aggregation from pairwise data, and show that this converges to an optimal ranking under an even more general condition that we term 'generalized low-noise'. In all cases, we provide explicit sample complexity bounds for exact recovery of an optimal ranking. Our experiments confirm our theoretical findings and help to shed light on the statistical behavior of various rank aggregation algorithms. There are various parametric models for analyzing pairwise comparison data, including the Bradley-Terry-Luce (BTL) and Thurstone models, but their reliance on strong parametric assumptions is limiting. In this work, we study a flexible model for pairwise comparisons, under which the probabilities of outcomes are required only to satisfy a natural form of stochastic transitivity. This class includes parametric models including the BTL and Thurstone models as special cases, but is considerably more general. We provide various examples of models in this broader stochastically transitive class for which classical parametric models provide poor fits. Despite this greater flexibility, we show that the matrix of probabilities can be estimated at the same rate as in standard parametric models. On the other hand, unlike in the BTL and Thurstone models, computing the minimax-optimal estimator in the stochastically transitive model is non-trivial, and we explore various computationally tractable alternatives. We show that a simple singular value thresholding algorithm is statistically consistent but does not achieve the minimax rate. We then propose and study algorithms that achieve the minimax rate over interesting sub-classes of the full stochastically transitive class. We complement our theoretical results with thorough numerical simulations. In this paper we present a general treatment of the preference aggregation problem, in which multiple preferences over objects must be combined into a single consensus ranking. We consider two instances of this problem: unsupervised aggregation where no information about a target ranking is available, and supervised aggregation where ground truth preferences are provided. For each problem class we develop novel learning methods that are applicable to a wide range of preference types. Specifically, for unsupervised aggregation we introduce the Multinomial Preference model (MPM) which uses a multinomial generative process to model the observed preferences. For the supervised problem we develop a supervised extension for MPM and then propose two fully supervised models. The first model employs SVD factorization to derive effective item features, transforming the aggregation problems into a learning-to-rank one. The second model aims to eliminate the costly SVD factorization and instantiates a probabilistic CRF framework, deriving unary and pairwise potentials directly from the observed preferences. Using a probabilistic framework allows us to directly optimize the expectation of any target metric, such as NDCG or ERR. All the proposed models operate on pairwise preferences and can thus be applied to a wide range of preference types. We empirically validate the models on rank aggregation and collaborative filtering data sets and demonstrate superior empirical accuracy. This paper is concerned with rank aggregation, which aims to combine multiple input rankings to get a better ranking. A popular approach to rank aggregation is based on probabilistic models on permutations, e.g., the Luce model and the Mallows model. However, these models have their limitations in either poor expressiveness or high computational complexity. To avoid these limitations, in this paper, we propose a new model, which is defined with a coset-permutation distance, and models the generation of a permutation as a stagewise process. We refer to the new model as coset-permutation distance based stagewise (CPS) model. The CPS model has rich expressiveness and can therefore be used in versatile applications, because many different permutation distances can be used to induce the coset-permutation distance. The complexity of the CPS model is low because of the stagewise decomposition of the permutation probability and the efficient computation of most coset-permutation distances. We apply the CPS model to supervised rank aggregation, derive the learning and inference algorithms, and empirically study their effectiveness and efficiency. Experiments on public datasets show that the derived algorithms based on the CPS model can achieve state-of-the-art ranking accuracy, and are much more efficient than previous algorithms. Given a set V of n elements we wish to linearly order them given pairwise preference labels which may be non-transitive (due to irrationality or arbitrary noise). The goal is to linearly order the elements while disagreeing with as few pairwise preference labels as possible. Our performance is measured by two parameters: The number of disagreements (loss) and the query complexity (number of pairwise preference labels). Our algorithm adaptively queries at most O(e-6n log5 n) preference labels for a regret of e times the optimal loss. As a function of n, this is asymptotically better than standard (non-adaptive) learning bounds achievable for the same problem. Our main result takes us a step closer toward settling an open problem posed by learning-to-rank (from pairwise information) theoreticians and practitioners: What is a provably correct way to sample preference labels? To further show the power and practicality of our solution, we analyze a typical test case in which a large margin linear relaxation is used for efficiently solving the simpler learning problems in our decomposition.