Active Ranking using Pairwise Comparisons

TL;DR

This paper explores efficient algorithms for ranking objects based on pairwise comparisons when objects are embedded in a Euclidean space, showing that fewer comparisons are needed under certain geometric assumptions.

Contribution

It introduces a novel algorithm that identifies rankings with significantly fewer comparisons in Euclidean space embeddings, improving efficiency over standard methods.

Findings

Number of rankings grows as n^{2d} under Euclidean embedding

Algorithm identifies a ranking with about d log n comparisons on average

Most comparisons are needed if comparisons are chosen at random

Abstract

This paper examines the problem of ranking a collection of objects using pairwise comparisons (rankings of two objects). In general, the ranking of $n$ objects can be identified by standard sorting methods using $n log_2 n$ pairwise comparisons. We are interested in natural situations in which relationships among the objects may allow for ranking using far fewer pairwise comparisons. Specifically, we assume that the objects can be embedded into a $d$-dimensional Euclidean space and that the rankings reflect their relative distances from a common reference point in $R^d$. We show that under this assumption the number of possible rankings grows like $n^{2d}$ and demonstrate an algorithm that can identify a randomly selected ranking using just slightly more than $d log n$ adaptively selected pairwise comparisons, on average. If instead the comparisons are chosen at random, then almost all pairwise comparisons must be made in order to identify any ranking. In addition, we propose a robust, error-tolerant algorithm that only requires that the pairwise comparisons are probably correct. Experimental studies with synthetic and real datasets support the conclusions of our theoretical analysis.