We consider spectral approaches to the problem of ranking n players given their incomplete and noisy pairwise comparisons, but revisit this classical problem in light of player covariate information. We propose three spectral ranking methods that incorporate player covariates and are based on seriation, low-rank structure assumption and canonical correlation, respectively. Extensive numerical simulations on both synthetic and real-world data sets demonstrated that our proposed methods compare favorably to existing state-of-the-art covariate-based ranking algorithms.
We consider a problem of ranking and selection via simulation in the context of personalized decision making, where the best alternative is not universal but varies as a function of some observable covariates. The goal of ranking and selection with covariates (R&S-C) is to use simulation samples to obtain a selection policy that specifies the best alternative with certain statistical guarantee for subsequent individuals upon observing their covariates. A linear model is proposed to capture the relationship between the mean performance of an alternative and the covariates. Under the indifferenc…
Permutation-valued features arise in a variety of applications, either in a direct way when preferences are elicited over a collection of items, or an indirect way in which numerical ratings are converted to a ranking. To date, there has been relatively limited study of regression, classification, and testing problems based on permutation-valued features, as opposed to permutation-valued responses. This paper studies the use of reproducing kernel Hilbert space methods for learning from permutation-valued features. These methods embed the rankings into an implicitly defined function space, and …
Department of Mathematics Hong Kong Baptist University（香港 Baptist 大学数学系）；School of Mathematical Sciences Zhejiang University（浙江大学数学系）；School of Mathematical Sciences and Shanghai Key Laboratory for Contemporary Applied Mathematics Fudan University（复旦大学数学系和上海当代应用数学重点实验室）
Among community detection methods, spectral clustering enjoys two desirable properties: computational efficiency and theoretical guarantees of consistency. Most studies of spectral clustering consider only the edges of a network as input to the algorithm. Here we consider the problem of performing community detection in the presence of discrete node covariates, where network structure is determined by a combination of a latent block model structure and homophily on the observed covariates. We propose a spectral algorithm that we prove achieves perfect clustering with high probability on a clas…
We describe a seriation algorithm for ranking a set of items given pairwise comparisons between these items. Intuitively, the algorithm assigns similar rankings to items that compare similarly with all others. It does so by constructing a similarity matrix from pairwise comparisons, using seriation methods to reorder this matrix and construct a ranking. We first show that this spectral seriation algorithm recovers the true ranking when all pairwise comparisons are observed and consistent with a total order. We then show that ranking reconstruction is still exact when some pairwise comparisons …
University of Wisconsin（威斯康星大学）；Johns Hopkins University（约翰斯·霍普金斯大学）
Biological and social systems consist of myriad interacting units. The interactions can be represented in the form of a graph or network. Measurements of these graphs can reveal the underlying structure of these interactions, which provides insight into the systems that generated the graphs. Moreover, in applications such as connectomics, social networks, and genomics, graph data are accompanied by contextualizing measures on each node. We utilize these node covariates to help uncover latent communities in a graph, using a modification of spectral clustering. Statistical guarantees are provide…
Google（谷歌）；University of Michigan（密歇根大学）；University of Washington（华盛顿大学）；Yale University（耶鲁大学）
Given a number of pairwise preferences of items, a common task is to rank all the items. Examples include pairwise movie ratings, New Yorker cartoon caption contests, and many other consumer preferences tasks. What these settings have in common is two-fold: a scarcity of data (it may be costly to get comparisons for all the pairs of items) and additional feature information about the items (e.g., movie genre, director, and cast). In this paper we modify a popular and well studied method, RankCentrality for rank aggregation to account for few comparisons and that incorporates additional feature…
Johns Hopkins University（约翰斯·霍普金斯大学）；University of Pittsburgh（匹兹堡大学）
In network inference applications, it is often desirable to detect community structure, namely to cluster vertices into groups, or blocks, according to some measure of similarity. Beyond mere adjacency matrices, many real networks also involve vertex covariates that carry key information about underlying block structure in graphs. To assess the effects of such covariates on block recovery, we present a comparative analysis of two model-based spectral algorithms for clustering vertices in stochastic blockmodel graphs with vertex covariates. The first algorithm uses only the adjacency matrix, an…
In many applications of network analysis, it is important to distinguish between observed and unobserved factors affecting network structure. To this end, we develop spectral estimators for both unobserved blocks and the effect of covariates in stochastic blockmodels. On the theoretical side, we establish asymptotic normality of our estimators for the subsequent purpose of performing inference. On the applied side, we show that computing our estimator is much faster than standard variational expectation--maximization algorithms and scales well for large networks. Monte Carlo experiments sugges…
This article introduces a flexible and adaptive nonparametric method for estimating the association between multiple covariates and power spectra of multiple time series. The proposed approach uses a Bayesian sum of trees model to capture complex dependencies and interactions between covariates and the power spectrum, which are often observed in studies of biomedical time series. Local power spectra corresponding to terminal nodes within trees are estimated nonparametrically using Bayesian penalized linear splines. The trees are considered to be random and fit using a Bayesian backfitting Mark…
KAIST（韩国科学技术院）；Electronics and Telecommunications Research Institute（电子与电信研究所）；UIUC（伊利诺伊大学厄巴纳-香槟分校）
We explore the top-$K$ rank aggregation problem. Suppose a collection of items is compared in pairs repeatedly, and we aim to recover a consistent ordering that focuses on the top-$K$ ranked items based on partially revealed preference information. We investigate the Bradley-Terry-Luce model in which one ranks items according to their perceived utilities modeled as noisy observations of their underlying true utilities. Our main contributions are two-fold. First, in a general comparison model where item pairs to compare are given a priori, we attain an upper and lower bound on the sample size f…
North Carolina State University（北卡罗来纳州立大学）；London School of Economics and Political Science（伦敦政治经济学院）
Dynamic treatment regimes assign personalized treatments to patients sequentially over time based on their baseline information and time-varying covariates. In mobile health applications, these covariates are typically collected at different frequencies over a long time horizon. In this paper, we propose a deep spectral Q-learning algorithm, which integrates principal component analysis (PCA) with deep Q-learning to handle the mixed frequency data. In theory, we prove that the mean return under the estimated optimal policy converges to that under the optimal one and establish its rate of conve…
As larger and more comprehensive datasets become standard in contemporary machine learning, it becomes increasingly more difficult to obtain reliable, trustworthy label information with which to train sophisticated models. To address this problem, crowdsourcing has emerged as a popular, inexpensive, and efficient data mining solution for performing distributed label collection. However, crowdsourced annotations are inherently untrustworthy, as the labels are provided by anonymous volunteers who may have varying, unreliable expertise. Worse yet, some participants on commonly used platforms such…
Given partially observed pairwise comparison data generated by the Bradley-Terry-Luce (BTL) model, we study the problem of top-$k$ ranking. That is, to optimally identify the set of top-$k$ players. We derive the minimax rate with respect to a normalized Hamming loss. This provides the first result in the literature that characterizes the partial recovery error in terms of the proportion of mistakes for top-$k$ ranking. We also derive the optimal signal to noise ratio condition for the exact recovery of the top-$k$ set. The maximum likelihood estimator (MLE) is shown to achieve both optimal pa…
We consider the classic problem of establishing a statistical ranking of a set of n items given a set of inconsistent and incomplete pairwise comparisons between such items. Instantiations of this problem occur in numerous applications in data analysis (e.g., ranking teams in sports data), computer vision, and machine learning. We formulate the above problem of ranking with incomplete noisy information as an instance of the group synchronization problem over the group SO(2) of planar rotations, whose usefulness has been demonstrated in numerous applications in recent years. Its least squares s…
This paper is concerned with the problem of top-$K$ ranking from pairwise comparisons. Given a collection of $n$ items and a few pairwise comparisons across them, one wishes to identify the set of $K$ items that receive the highest ranks. To tackle this problem, we adopt the logistic parametric model --- the Bradley-Terry-Luce model, where each item is assigned a latent preference score, and where the outcome of each pairwise comparison depends solely on the relative scores of the two items involved. Recent works have made significant progress towards characterizing the performance (e.g. the m…
Improved theoretical guarantee for rank aggregation via spectral method
Shanghai Frontiers Science Center of Artificial Intelligence and Deep Learning（上海人工智能与深度学习前沿科学中心）；New York University Shanghai（上海纽约大学）；Center for Data Science, New York University（纽约大学数据科学中心）；New York University（纽约大学）
University of Utah（犹他大学）；University of Twente（特文特大学）；University of California（加利福尼亚大学）
Given a graph where vertices represent alternatives and arcs represent pairwise comparison data, the statistical ranking problem is to find a potential function, defined on the vertices, such that the gradient of the potential function agrees with the pairwise comparisons. Our goal in this paper is to develop a method for collecting data for which the least squares estimator for the ranking problem has maximal Fisher information. Our approach, based on experimental design, is to view data collection as a bi-level optimization problem where the inner problem is the ranking problem and the outer…
The optimal design of experiments typically involves solving an NP-hard combinatorial optimization problem. In this paper, we aim to develop a globally convergent and practically efficient optimization algorithm. Specifically, we consider a setting where the pre-treatment outcome data is available and the synthetic control estimator is invoked. The average treatment effect is estimated via the difference between the weighted average outcomes of the treated and control units, where the weights are learned from the observed data. {Under this setting, we surprisingly observed that the optimal exp…
University of Chinese Academy of Sciences（中国科学院大学）；Institute of Computing Technology, Chinese Academy of Sciences（中国科学院计算技术研究所）；Jiangxi Normal University（江西师范大学）；Sun Yat-sen University（中山大学）；Peng Cheng Laboratory（鹏城实验室）
Rank aggregation with pairwise comparisons has shown promising results in elections, sports competitions, recommendations, and information retrieval. However, little attention has been paid to the security issue of such algorithms, in contrast to numerous research work on the computational and statistical characteristics. Driven by huge profits, the potential adversary has strong motivation and incentives to manipulate the ranking list. Meanwhile, the intrinsic vulnerability of the rank aggregation methods is not well studied in the literature. To fully understand the possible risks, we focus …
Ranked data sets, where m judges/voters specify a preference ranking of n objects/candidates, are increasingly prevalent in contexts such as political elections, computer vision, recommender systems, and bioinformatics. The vote counts for each ranking can be viewed as an n! data vector lying on the permutahedron, which is a Cayley graph of the symmetric group with vertices labeled by permutations and an edge when two permutations differ by an adjacent transposition. Leveraging combinatorial representation theory and recent progress in signal processing on graphs, we investigate a novel, scala…
