Optimal Instance Adaptive Algorithm for the Top- K Ranking Problem

• Published in: IEEE transactions on information theory Vol. 64; no. 9; pp. 6139 - 6160
• Main Authors: Xi Chen, Gopi, Sivakanth, Jieming Mao, Schneider, Jon
• Format: Journal Article
• Language: English
• Published: IEEE 01.09.2018
• Subjects: Adaptation models; Complexity theory; Computer science; Electronic mail; instance adaptive; Noise measurement; Parametric statistics; rank aggregation; Stochastic processes; strong stochastic transitivity; Top-<italic xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">K ranking
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2018.2851986

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 Õ(√n) 1 ; i.e., to solve any instance of top-K ranking, our algorithm needs at most Õ(√n) times as many samples needed as the best possible algorithm for that instance [in contrast, all previous known algorithms for the topK problem have competitive ratios of Ω̃(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 Ω̃(√n).