Lenient Regret and Good-Action Identification in Gaussian Process Bandits

Xu Cai, Selwyn Gomes, Jonathan Scarlett
Proceedings of the 38th International Conference on Machine Learning, PMLR 139:1183-1192, 2021.

Abstract

In this paper, we study the problem of Gaussian process (GP) bandits under relaxed optimization criteria stating that any function value above a certain threshold is “good enough”. On the theoretical side, we study various {\em lenient regret} notions in which all near-optimal actions incur zero penalty, and provide upper bounds on the lenient regret for GP-UCB and an elimination algorithm, circumventing the usual O(√T) term (with time horizon T) resulting from zooming extremely close towards the function maximum. In addition, we complement these upper bounds with algorithm-independent lower bounds. On the practical side, we consider the problem of finding a single “good action” according to a known pre-specified threshold, and introduce several good-action identification algorithms that exploit knowledge of the threshold. We experimentally find that such algorithms can typically find a good action faster than standard optimization-based approaches.

Cite this Paper

BibTeX
@InProceedings{pmlr-v139-cai21c, title = {Lenient Regret and Good-Action Identification in Gaussian Process Bandits}, author = {Cai, Xu and Gomes, Selwyn and Scarlett, Jonathan}, booktitle = {Proceedings of the 38th International Conference on Machine Learning}, pages = {1183--1192}, year = {2021}, editor = {Meila, Marina and Zhang, Tong}, volume = {139}, series = {Proceedings of Machine Learning Research}, month = {18--24 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v139/cai21c/cai21c.pdf}, url = {https://proceedings.mlr.press/v139/cai21c.html}, abstract = {In this paper, we study the problem of Gaussian process (GP) bandits under relaxed optimization criteria stating that any function value above a certain threshold is “good enough”. On the theoretical side, we study various {\em lenient regret} notions in which all near-optimal actions incur zero penalty, and provide upper bounds on the lenient regret for GP-UCB and an elimination algorithm, circumventing the usual $O(\sqrt{T})$ term (with time horizon $T$) resulting from zooming extremely close towards the function maximum. In addition, we complement these upper bounds with algorithm-independent lower bounds. On the practical side, we consider the problem of finding a single “good action” according to a known pre-specified threshold, and introduce several good-action identification algorithms that exploit knowledge of the threshold. We experimentally find that such algorithms can typically find a good action faster than standard optimization-based approaches.} }
Endnote
%0 Conference Paper %T Lenient Regret and Good-Action Identification in Gaussian Process Bandits %A Xu Cai %A Selwyn Gomes %A Jonathan Scarlett %B Proceedings of the 38th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2021 %E Marina Meila %E Tong Zhang %F pmlr-v139-cai21c %I PMLR %P 1183--1192 %U https://proceedings.mlr.press/v139/cai21c.html %V 139 %X In this paper, we study the problem of Gaussian process (GP) bandits under relaxed optimization criteria stating that any function value above a certain threshold is “good enough”. On the theoretical side, we study various {\em lenient regret} notions in which all near-optimal actions incur zero penalty, and provide upper bounds on the lenient regret for GP-UCB and an elimination algorithm, circumventing the usual $O(\sqrt{T})$ term (with time horizon $T$) resulting from zooming extremely close towards the function maximum. In addition, we complement these upper bounds with algorithm-independent lower bounds. On the practical side, we consider the problem of finding a single “good action” according to a known pre-specified threshold, and introduce several good-action identification algorithms that exploit knowledge of the threshold. We experimentally find that such algorithms can typically find a good action faster than standard optimization-based approaches.
APA
Cai, X., Gomes, S. & Scarlett, J.. (2021). Lenient Regret and Good-Action Identification in Gaussian Process Bandits. Proceedings of the 38th International Conference on Machine Learning, in Proceedings of Machine Learning Research 139:1183-1192 Available from https://proceedings.mlr.press/v139/cai21c.html.

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