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This paper explores the preference-based top-$K$ rank aggregation problem.Suppose that a collection of items is repeatedly compared in pairs, and onewishes to recover a consistent ordering that emphasizes the top-$K$ rankeditems, based on partially revealed preferences. We focus on theBradley-Terry-Luce (BTL) model that postulates a set of latent preferencescores underlying all items, where the odds of paired comparisons depend onlyon the relative scores of the items involved.  We characterize the minimax limits on identifiability of top-$K$ rankeditems, in the presence of random and non-adaptive sampling. Our resultshighlight a separation measure that quantifies the gap of preference scoresbetween the $K^{\text{th}}$ and $(K+1)^{\text{th}}$ ranked items. The minimumsample complexity required for reliable top-$K$ ranking scales inversely withthe separation measure irrespective of other preference distribution metrics.To approach this minimax limit, we propose a nearly linear-time ranking scheme,called \emph{Spectral MLE}, that returns the indices of the top-$K$ items inaccordance to a careful score estimate. In a nutshell, Spectral MLE starts withan initial score estimate with minimal squared loss (obtained via a spectralmethod), and then successively refines each component with the assistance ofcoordinate-wise MLEs. Encouragingly, Spectral MLE allows perfect top-$K$ itemidentification under minimal sample complexity. The practical applicability ofSpectral MLE is further corroborated by numerical experiments.

Spectral MLE: Top-$K$ Rank Aggregation from Pairwise Comparisons