Maximum Selection and Ranking under Noisy Comparisons

We consider $(ε,δ)$-PAC maximum-selection and ranking for general probabilistic models whose comparisons probabilities satisfy strong stochastic transitivity and stochastic triangle inequality. Modifying the popular knockout tournament, we propose a maximum-selection algorithm that uses $\mathcal{O}\left(\frac{n}{ε^2}\log \frac{1}δ\right)$ comparisons, a number tight up to a constant factor. We then derive a general framework that improves the performance of many ranking algorithms, and combine it with merge sort and binary search to obtain a ranking algorithm that uses $\mathcal{O}\left(\frac{n\log n (\log \log n)^3}{ε^2}\right)$ comparisons for any $δ\ge\frac1n$, a number optimal up to a $(\log \log n)^3$ factor.