Minimax Rate for Learning From Pairwise Comparisons in the BTL Model

Julien Hendrickx, Alex Olshevsky, Venkatesh Saligrama
Proceedings of the 37th International Conference on Machine Learning, PMLR 119:4193-4202, 2020.

Abstract

We consider the problem of learning the qualities w_1, ... , w_n of a collection of items by performing noisy comparisons among them. We assume there is a fixed “comparison graph” and every neighboring pair of items is compared k times. We will study the popular Bradley-Terry-Luce model, where the probability that item i wins a comparison against j equals w_i/(w_i + w_j). We are interested in how the expected error in estimating the vector w = (w_1, ... , w_n) behaves in the regime when the number of comparisons k is large. Our contribution is the determination of the minimax rate up to a constant factor. We show that this rate is achieved by a simple algorithm based on weighted least squares, with weights determined from the empirical outcomes of the comparisons. This algorithm can be implemented in nearly linear time in the total number of comparisons.

Cite this Paper

BibTeX
@InProceedings{pmlr-v119-hendrickx20a, title = {Minimax Rate for Learning From Pairwise Comparisons in the {BTL} Model}, author = {Hendrickx, Julien and Olshevsky, Alex and Saligrama, Venkatesh}, booktitle = {Proceedings of the 37th International Conference on Machine Learning}, pages = {4193--4202}, year = {2020}, editor = {III, Hal Daumé and Singh, Aarti}, volume = {119}, series = {Proceedings of Machine Learning Research}, month = {13--18 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v119/hendrickx20a/hendrickx20a.pdf}, url = {https://proceedings.mlr.press/v119/hendrickx20a.html}, abstract = {We consider the problem of learning the qualities w_1, ... , w_n of a collection of items by performing noisy comparisons among them. We assume there is a fixed “comparison graph” and every neighboring pair of items is compared k times. We will study the popular Bradley-Terry-Luce model, where the probability that item i wins a comparison against j equals w_i/(w_i + w_j). We are interested in how the expected error in estimating the vector w = (w_1, ... , w_n) behaves in the regime when the number of comparisons k is large. Our contribution is the determination of the minimax rate up to a constant factor. We show that this rate is achieved by a simple algorithm based on weighted least squares, with weights determined from the empirical outcomes of the comparisons. This algorithm can be implemented in nearly linear time in the total number of comparisons.} }
Endnote
%0 Conference Paper %T Minimax Rate for Learning From Pairwise Comparisons in the BTL Model %A Julien Hendrickx %A Alex Olshevsky %A Venkatesh Saligrama %B Proceedings of the 37th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2020 %E Hal Daumé III %E Aarti Singh %F pmlr-v119-hendrickx20a %I PMLR %P 4193--4202 %U https://proceedings.mlr.press/v119/hendrickx20a.html %V 119 %X We consider the problem of learning the qualities w_1, ... , w_n of a collection of items by performing noisy comparisons among them. We assume there is a fixed “comparison graph” and every neighboring pair of items is compared k times. We will study the popular Bradley-Terry-Luce model, where the probability that item i wins a comparison against j equals w_i/(w_i + w_j). We are interested in how the expected error in estimating the vector w = (w_1, ... , w_n) behaves in the regime when the number of comparisons k is large. Our contribution is the determination of the minimax rate up to a constant factor. We show that this rate is achieved by a simple algorithm based on weighted least squares, with weights determined from the empirical outcomes of the comparisons. This algorithm can be implemented in nearly linear time in the total number of comparisons.
APA
Hendrickx, J., Olshevsky, A. & Saligrama, V.. (2020). Minimax Rate for Learning From Pairwise Comparisons in the BTL Model. Proceedings of the 37th International Conference on Machine Learning, in Proceedings of Machine Learning Research 119:4193-4202 Available from https://proceedings.mlr.press/v119/hendrickx20a.html.

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