The problem of rank aggregation from pairwise and multiway comparisons has a wide range of implications, ranging from recommendation systems to sports rankings to social choice. Some of the most popular algorithms for this problem come from the class of spectral ranking algorithms; these include the rank centrality (RC) algorithm for pairwise comparisons, which returns consistent estimates under the Bradley-Terry-Luce (BTL) model for pairwise comparisons (Negahban et al., 2017), and its generalization, the Luce spectral ranking (LSR) algorithm, which returns consistent estimates under the more general multinomial logit (MNL) model for multiway comparisons (Maystre & Grossglauser, 2015). In this paper, we design a provably faster spectral ranking algorithm, which we call accelerated spectral ranking (ASR), that is also consistent under the MNL/BTL models.Our accelerated algorithm is achieved by designing a random walk that has a faster mixing time than the random walks associated with previous algorithms. In addition to a faster algorithm, our results yield improved sample complexity bounds for recovery of the MNL/BTL parameters: to the best of our knowledge, we give the first general sample complexity bounds for recovering the parameters of the MNL model from multiway comparisons under any (connected) comparison graph (and improve significantly over previous bounds for the BTL model for pairwise comparisons). We also give a message-passing interpretation of our algorithm, which suggests a decentralized distributed implementation. Our experiments on several real-world and synthetic datasets confirm that our new ASR algorithm is indeed orders of magnitude faster than existing algorithms.

We propose a novel and flexible rank-breaking-then-composite-marginal-likelihood (RBCML) framework for learning random utility models (RUMs), which include the Plackett-Luce model. We characterize conditions for the objective function of RBCML to be strictly log-concave by proving that strict log-concavity is preserved under convolution and marginalization. We characterize necessary and sufficient conditions for RBCML to satisfy consistency and asymptotic normality. Experiments on synthetic data show that RBCML for Gaussian RUMs achieves better statistical efficiency and computation efficiency than the state-of-the-art algorithm and our RBCML for the Plackett-Luce model provides flexible tradeoffs between running time and statistical efficiency.

We propose a new statistical model for ranking data, i.e., a new family of probability distributions on permutations. Our model is inspired by the idea of a data-generating process in the form of a noisy sorting procedure, in which deterministic comparisons between pairs of items are replaced by Bernoulli trials. The probability of producing a certain ranking as a result then essentially depends on the Bernoulli parameters, which can be interpreted as pairwise preferences. We show that our model can be written in closed form if insertion or quick sort are used as sorting algorithms, and propose a maximum likelihood approach for parameter estimation. We also introduce a generalization of the model, in which the constraints on pairwise preferences are relaxed, and for which maximum likelihood estimation can be carried out based on a variation of the generalized iterative scaling algorithm. Experimentally, we show that the models perform very well in terms of goodness of fit, compared to existing models for ranking data.

In this paper, we propose a listwise approach for constructing user-specific rankings in recommendation systems in a collaborative fashion. We contrast the listwise approach to previous pointwise and pairwise approaches, which are based on treating either each rating or each pairwise comparison as an independent instance respectively. By extending the work of ListNet (Cao et al., 2007), we cast listwise collaborative ranking as maximum likelihood under a permutation model which applies probability mass to permutations based on a low rank latent score matrix. We present a novel algorithm called SQL-Rank, which can accommodate ties and missing data and can run in linear time. We develop a theoretical framework for analyzing listwise ranking methods based on a novel representation theory for the permutation model. Applying this framework to collaborative ranking, we derive asymptotic statistical rates as the number of users and items grow together. We conclude by demonstrating that our SQL-Rank method often outperforms current state-of-the-art algorithms for implicit feedback such as Weighted-MF and BPR and achieve favorable results when compared to explicit feedback algorithms such as matrix factorization and collaborative ranking.

We consider the setting where we wish to perform ranking for hundreds of thousands of users which is common in recommender systems and web search ranking. Learning a single ranking function is unlikely to capture the variability across all users while learning a ranking function for each person is time-consuming and requires large amounts of data from each user. To address this situation, we propose a Factorization RankSVM algorithm which learns a series of k basic ranking functions and then constructs for each user a local ranking function that is a combination of them. We develop a fast algorithm to reduce the time complexity of gradient descent solver by exploiting the low-rank structure, and the resulting algorithm is much faster than existing methods. Furthermore, we prove that the generalization error of the proposed method can be significantly better than training individual RankSVMs. Finally, we present some interesting patterns in the principal ranking functions learned by our algorithms.