Maximum Selection and Ranking under Noisy Comparisons
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Proceedings of the 34th International Conference on Machine Learning, PMLR 70:10881096, 2017.
Abstract
We consider $(\epsilon,\delta)$PAC maximumselection and ranking using pairwise comparisons for general probabilistic models whose comparison probabilities satisfy strong stochastic transitivity and stochastic triangle inequality. Modifying the popular knockout tournament, we propose a simple maximumselection 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.
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