On the Sample Complexity of Rank Regression from Pairwise Comparisons

TL;DR

This paper analyzes the sample complexity of learning rank regression models from pairwise comparisons, establishing bounds on the number of comparisons needed for accurate rank estimation in high-dimensional settings.

Contribution

It provides theoretical bounds on the number of pairwise comparisons required to accurately learn rank regression models, linking sample complexity to feature dimension and dataset size.

Findings

Comparison complexity scales with dimension and dataset size.

Accurate rank regression is achievable with a logarithmic number of comparisons.

Theoretical bounds guide practical data collection for ranking tasks.

Abstract

We consider a rank regression setting, in which a dataset of $N$ samples with features in $\mathbb{R}^d$ is ranked by an oracle via $M$ pairwise comparisons. Specifically, there exists a latent total ordering of the samples; when presented with a pair of samples, a noisy oracle identifies the one ranked higher with respect to the underlying total ordering. A learner observes a dataset of such comparisons and wishes to regress sample ranks from their features. We show that to learn the model parameters with $\epsilon > 0$ accuracy, it suffices to conduct $M \in \Omega(dN\log^3 N/\epsilon^2)$ comparisons uniformly at random when $N$ is $\Omega(d/\epsilon^2)$.