Maximum Selection and Ranking under Noisy Comparisons

We consider $(\epsilon,\delta)$-PAC maximum-selection and ranking using pairwise comparisons for \nobreak{general} probabilistic models whose comparison probabilities satisfy strong stochastic transitivity and stochastic triangle inequality. Modifying the popular knockout tournament, we propose a simple maximum-selection algorithm that uses $\mathcal{O}\left(\frac{n}{\epsilon^2} \left(1+\log \frac1{\delta}\right)\right)$ comparisons, optimal up to a constant factor. We then derive a general framework that uses noisy binary search to speed up many ranking algorithms, and combine it with merge sort to obtain a ranking algorithm that uses $\mathcal{O}\left(\frac n{\epsilon^2}\log n(\log \log n)^3\right)$ comparisons for $\delta=\frac1n$, optimal up to a $(\log \log n)^3$ factor.