Optimal Instance Adaptive Algorithm for the Top-$K$ Ranking Problem

Abstract

Motivated by applications in recommender systems, web search, social choice, and crowdsourcing, we consider the problem of identifying the set of top $K$ items from noisy pairwise comparisons. In our setting, we are given $r$ pairwise comparisons between each pair of $n$ items, where each comparison has noise constrained by a very general noise model called the strong stochastic transitivity model. Our goal is to provide an optimal instance adaptive algorithm for the top- $K$ ranking problem. In particular, we present a linear time algorithm that has a competitive ratio of $\tilde {O}(\sqrt {n})$ 1 ; i.e., to solve any instance of top- $K$ ranking, our algorithm needs at most $\tilde {O}(\sqrt {n})$ times as many samples needed as the best possible algorithm for that instance [in contrast, all previous known algorithms for the top- $K$ problem have competitive ratios of $\tilde {\Omega }(n)$ or worse]. We further show that this is tight (up to polylogarithmic factors): any algorithm for the top- $K$ problem has competitive ratio of at least $\tilde {\Omega }(\sqrt {n})$ . 1 We use $\tilde {O}$ and $\tilde {\Omega }$ notation to hide polylogarithmic factors.