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We explore the top-$K$ rank aggregation problem. Suppose a collection ofitems is compared in pairs repeatedly, and we aim to recover a consistentordering that focuses on the top-$K$ ranked items based on partially revealedpreference information. We investigate the Bradley-Terry-Luce model in whichone ranks items according to their perceived utilities modeled as noisyobservations of their underlying true utilities. Our main contributions aretwo-fold. First, in a general comparison model where item pairs to compare aregiven a priori, we attain an upper and lower bound on the sample size forreliable recovery of the top-$K$ ranked items. Second, more importantly,extending the result to a random comparison model where item pairs to compareare chosen independently with some probability, we show that in slightlyrestricted regimes, the gap between the derived bounds reduces to a constantfactor, hence reveals that a spectral method can achieve the minimax optimalityon the (order-wise) sample size required for top-$K$ ranking. That is to say,we demonstrate a spectral method alone to be sufficient to achieve theoptimality and advantageous in terms of computational complexity, as it doesnot require an additional stage of maximum likelihood estimation that astate-of-the-art scheme employs to achieve the optimality. We corroborate ourmain results by numerical experiments.

Top-$K$ Ranking from Pairwise Comparisons: When Spectral Ranking is Optimal