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This work 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 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 Kth and (K + 1)th ranked items. The minimum sample complexity required for reliable top-K ranking scales inversely with the separation measure. To approach this minimax limit, we propose a nearly linear-time ranking scheme that returns the indices of the top K items in accordance to a careful score estimate.
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