Spectral Method and Regularized MLE Are Both Optimal for Top-$K$ Ranking

Abstract

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 mean square error for estimating the scores) of several classical methods, including the spectral method and the maximum likelihood estimator (MLE). However, where they stand regarding top-$K$ ranking remains unsettled. We demonstrate that under a natural random sampling model, the spectral method alone, or the regularized MLE alone, is minimax optimal in terms of the sample complexity --- the number of paired comparisons needed to ensure exact top-$K$ identification, for the fixed dynamic range regime. This is accomplished via optimal control of the entrywise error of the score estimates. We complement our theoretical studies by numerical experiments, confirming that both methods yield low entrywise errors for estimating the underlying scores. Our theory is established via a novel leave-one-out trick, which proves effective for analyzing both iterative and non-iterative procedures. Along the way, we derive an elementary eigenvector perturbation bound for probability transition matrices, which parallels the Davis-Kahan $\sin \Theta$ theorem for symmetric matrices. This also allows us to close the gap between the ${\ell }_2$ error upper bound for the spectral method and the minimax lower limit.

Add discussions on the setting of the general condition number

References in corpus (7)

• Implicit Regularization in Nonconvex Statistical Estimation: Gradient Descent Converges Linearly for Phase Retrieval, Matrix Completion, and Blind Deconvolution
• Near-optimal bounds for phase synchronization
• Spectral MLE: Top-$K$ Rank Aggregation from Pairwise Comparisons
• Entrywise Eigenvector Analysis of Random Matrices with Low Expected Rank
• Unperturbed: spectral analysis beyond Davis-Kahan
• Worst-case vs Average-case Design for Estimation from Fixed Pairwise Comparisons
• Individualized Rank Aggregation using Nuclear Norm Regularization

Cited by in corpus (10)

• Nonconvex Optimization Meets Low-Rank Matrix Factorization: An Overview
• Implicit Regularization in Nonconvex Statistical Estimation: Gradient Descent Converges Linearly for Phase Retrieval, Matrix Completion, and Blind Deconvolution
• Lagrangian Inference for Ranking Problems
• Iterative Collaborative Filtering for Sparse Matrix Estimation
• A Tale of HodgeRank and Spectral Method: Target Attack Against Rank Aggregation Is the Fixed Point of Adversarial Game
• Targeted sampling from massive block model graphs with personalized PageRank
• Optimal tuning-free convex relaxation for noisy matrix completion
• Sequential Manipulation Against Rank Aggregation: Theory and Algorithm
• Minimax Hypothesis Testing for the Bradley-Terry-Luce Model
• Robust Max Entrywise Error Bounds for Tensor Estimation from Sparse Observations via Similarity Based Collaborative Filtering