On the Sample Complexity of Rank Regression from Pairwise Comparisons

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 $\Omega (d/{\epsilon }^2)$.

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