Competitive analysis of the top-K ranking problem

TL;DR

This paper introduces a linear-time algorithm for the top-$K$ ranking problem under noisy pairwise comparisons with a strong stochastic transitivity model, achieving near-optimal competitive ratio and improving over prior methods.

Contribution

It presents a new linear-time algorithm with a competitive ratio of  O(\u007E( )), matching the lower bound and advancing the efficiency of top-$K$ ranking under noise.

Findings

The algorithm has a competitive ratio of  O(( ))

The lower bound on competitive ratio is  O(( ))

The method is applicable to noisy comparison settings in recommender systems and search

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 non-actively 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 (SST) model. We analyze the competitive ratio of algorithms for the top-$K$ problem. In particular, we present a linear time algorithm for the top-$K$ problem which has a competitive ratio of $\tilde{O}(\sqrt{n})$; i.e. to solve any instance of top-$K$, 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: any algorithm for the top-$K$ problem has competitive ratio at least $\tilde{\Omega}(\sqrt{n})$.