  This paper explores the preference-based top-$K$ rank aggregation problem.
Suppose that a collection of items is repeatedly compared in pairs, and one
wishes to recover a consistent ordering that emphasizes the top-$K$ ranked
items, based on partially revealed preferences. We focus on the
Bradley-Terry-Luce (BTL) model that postulates a set of latent preference
scores underlying all items, where the odds of paired comparisons depend only
on the relative scores of the items involved.
  We characterize the minimax limits on identifiability of top-$K$ ranked
items, in the presence of random and non-adaptive sampling. Our results
highlight a separation measure that quantifies the gap of preference scores
between the $K^{\text{th}}$ and $(K+1)^{\text{th}}$ ranked items. The minimum
sample complexity required for reliable top-$K$ ranking scales inversely with
the 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 in
accordance to a careful score estimate. In a nutshell, Spectral MLE starts with
an initial score estimate with minimal squared loss (obtained via a spectral
method), and then successively refines each component with the assistance of
coordinate-wise MLEs. Encouragingly, Spectral MLE allows perfect top-$K$ item
identification under minimal sample complexity. The practical applicability of
Spectral MLE is further corroborated by numerical experiments.
