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

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.Comment: Add discussions on the setting of the general condition numbe

Top-k selection with pairwise comparisons

In this work we consider active, pairwise top- selection, the problem of identifying the highest quality subset of given size from a set of alternatives, based on the information collected from noisy, sequentially chosen pairwise comparisons. We adapt two well known Bayesian sequential sampling techniques, the Knowledge Gradient policy and the Optimal Computing Budget Allocation framework for the pairwise setting and compare their performance on a range of empirical tests. We demonstrate that these methods are able to match or outperform the current state of the art racing algorithm approach

Efficient pairwise information collection for subset selection

In this work, we consider the problems of selecting the subset of the top-k best of a set of alternatives, where the fitness of alternatives must be estimated through noisy pairwise sampling. To do this, we propose two novel active pairwise sampling methods, adapted from popular non-pairwise ranking and selection frameworks. We prove that our proposed methods have desirable asymptotic properties, and demonstrate empirically that they can perform better than current state-of-the art pairwise selection algorithms on a range of tasks. We show how our proposed methods can be integrated into the Covariance Matrix Adaptation Evolutionary Strategy, to improve fitness evaluation and optimizer performance including in the evolution of neural network based agents for playing No Limit Texas Hold’em poker. Finally, we demonstrate how parametric models can be used to help our proposed sampling algorithms exploit transitive preference structure between alternative pairs