Efficient Kernelized UCB for Contextual Bandits
Houssam Zenati, Alberto Bietti, Eustache Diemert, Julien Mairal, Matthieu Martin, Pierre Gaillard
Abstract
In this paper, we tackle the computational efficiency of kernelized UCB algorithms in contextual bandits. While standard methods require a $\mathcal{O}(CT^3)$ complexity where $T$ is the horizon and the constant $C$ is related to optimizing the UCB rule, we propose an efficient contextual algorithm for large-scale problems. Specifically, our method relies on incremental Nyström approximations of the joint kernel embedding of contexts and actions. This allows us to achieve a complexity of $\mathcal{O}(CTm^2)$ where $m$ is the number of Nyström points. To recover the same regret as the standard kernelized UCB algorithm, $m$ needs to be of order of the effective dimension of the problem, which is at most $\mathcal{O}(\sqrt{T})$ and nearly constant in some cases.
BibTeX
@InProceedings{pmlr-v151-zenati22a,
title = { Efficient Kernelized UCB for Contextual Bandits },
author = {Zenati, Houssam and Bietti, Alberto and Diemert, Eustache and Mairal, Julien and Martin, Matthieu and Gaillard, Pierre},
booktitle = {Proceedings of The 25th International Conference on Artificial Intelligence and Statistics},
pages = {5689--5720},
year = {2022},
editor = {Camps-Valls, Gustau and Ruiz, Francisco J. R. and Valera, Isabel},
volume = {151},
series = {Proceedings of Machine Learning Research},
month = {28--30 Mar},
publisher = {PMLR},
pdf = {https://proceedings.mlr.press/v151/zenati22a/zenati22a.pdf},
url = {https://proceedings.mlr.press/v151/zenati22a.html},
abstract = { In this paper, we tackle the computational efficiency of kernelized UCB algorithms in contextual bandits. While standard methods require a $\mathcal{O}(CT^3)$ complexity where $T$ is the horizon and the constant $C$ is related to optimizing the UCB rule, we propose an efficient contextual algorithm for large-scale problems. Specifically, our method relies on incremental Nyström approximations of the joint kernel embedding of contexts and actions. This allows us to achieve a complexity of $\mathcal{O}(CTm^2)$ where $m$ is the number of Nyström points. To recover the same regret as the standard kernelized UCB algorithm, $m$ needs to be of order of the effective dimension of the problem, which is at most $\mathcal{O}(\sqrt{T})$ and nearly constant in some cases. }
}