Maximum Selection and Ranking under Noisy Comparisons

Moein Falahatgar, Alon Orlitsky, Venkatadheeraj Pichapati, Ananda Theertha Suresh
Proceedings of the 34th International Conference on Machine Learning, PMLR 70:1088-1096, 2017.

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.

Cite this Paper

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.} }
Endnote
%0 Conference Paper %T Maximum Selection and Ranking under Noisy Comparisons %A Moein Falahatgar %A Alon Orlitsky %A Venkatadheeraj Pichapati %A Ananda Theertha Suresh %B Proceedings of the 34th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2017 %E Doina Precup %E Yee Whye Teh %F pmlr-v70-falahatgar17a %I PMLR %P 1088--1096 %U https://proceedings.mlr.press/v70/falahatgar17a.html %V 70 %X 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.
APA
Falahatgar, M., Orlitsky, A., Pichapati, V. & Suresh, A.T.. (2017). Maximum Selection and Ranking under Noisy Comparisons. Proceedings of the 34th International Conference on Machine Learning, in Proceedings of Machine Learning Research 70:1088-1096 Available from https://proceedings.mlr.press/v70/falahatgar17a.html.

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