Maximum Selection and Ranking under Noisy Comparisons
Moein Falahatgar, Alon Orlitsky, Venkatadheeraj Pichapati, Ananda Theertha Suresh
Abstract
We consider $(\epsilon,\delta)$-PAC maximum-selection 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 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.
BibTeX
@InProceedings{pmlr-v70-falahatgar17a,
title = {Maximum Selection and Ranking under Noisy Comparisons},
author = {Moein Falahatgar and Alon Orlitsky and Venkatadheeraj Pichapati and Ananda Theertha Suresh},
booktitle = {Proceedings of the 34th International Conference on Machine Learning},
pages = {1088--1096},
year = {2017},
editor = {Precup, Doina and Teh, Yee Whye},
volume = {70},
series = {Proceedings of Machine Learning Research},
month = {06--11 Aug},
publisher = {PMLR},
pdf = {http://proceedings.mlr.press/v70/falahatgar17a/falahatgar17a.pdf},
url = {https://proceedings.mlr.press/v70/falahatgar17a.html},
abstract = {We consider $(\epsilon,\delta)$-PAC maximum-selection 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 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.}
}