Delayed Feedback in Kernel Bandits

Sattar Vakili, Danyal Ahmed, Alberto Bernacchia, Ciara Pike-Burke
Proceedings of the 40th International Conference on Machine Learning, PMLR 202:34779-34792, 2023.

Abstract

Black box optimisation of an unknown function from expensive and noisy evaluations is a ubiquitous problem in machine learning, academic research and industrial production. An abstraction of the problem can be formulated as a kernel based bandit problem (also known as Bayesian optimisation), where a learner aims at optimising a kernelized function through sequential noisy observations. The existing work predominantly assumes feedback is immediately available; an assumption which fails in many real world situations, including recommendation systems, clinical trials and hyperparameter tuning. We consider a kernel bandit problem under stochastically delayed feedback, and propose an algorithm with $\tilde{\mathcal{O}}\left(\sqrt{\Gamma_k(T) T}+\mathbb{E}[\tau]\right)$ regret, where $T$ is the number of time steps, $\Gamma_k(T)$ is the maximum information gain of the kernel with $T$ observations, and $\tau$ is the delay random variable. This represents a significant improvement over the state of the art regret bound of $\tilde{\mathcal{O}}\left(\Gamma_k(T)\sqrt{ T}+\mathbb{E}[\tau]\Gamma_k(T)\right)$ reported in (Verma et al., 2022). In particular, for very non-smooth kernels, the information gain grows almost linearly in time, trivializing the existing results. We also validate our theoretical results with simulations.

Cite this Paper

BibTeX
@InProceedings{pmlr-v202-vakili23a, title = {Delayed Feedback in Kernel Bandits}, author = {Vakili, Sattar and Ahmed, Danyal and Bernacchia, Alberto and Pike-Burke, Ciara}, booktitle = {Proceedings of the 40th International Conference on Machine Learning}, pages = {34779--34792}, year = {2023}, editor = {Krause, Andreas and Brunskill, Emma and Cho, Kyunghyun and Engelhardt, Barbara and Sabato, Sivan and Scarlett, Jonathan}, volume = {202}, series = {Proceedings of Machine Learning Research}, month = {23--29 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v202/vakili23a/vakili23a.pdf}, url = {https://proceedings.mlr.press/v202/vakili23a.html}, abstract = {Black box optimisation of an unknown function from expensive and noisy evaluations is a ubiquitous problem in machine learning, academic research and industrial production. An abstraction of the problem can be formulated as a kernel based bandit problem (also known as Bayesian optimisation), where a learner aims at optimising a kernelized function through sequential noisy observations. The existing work predominantly assumes feedback is immediately available; an assumption which fails in many real world situations, including recommendation systems, clinical trials and hyperparameter tuning. We consider a kernel bandit problem under stochastically delayed feedback, and propose an algorithm with $\tilde{\mathcal{O}}\left(\sqrt{\Gamma_k(T) T}+\mathbb{E}[\tau]\right)$ regret, where $T$ is the number of time steps, $\Gamma_k(T)$ is the maximum information gain of the kernel with $T$ observations, and $\tau$ is the delay random variable. This represents a significant improvement over the state of the art regret bound of $\tilde{\mathcal{O}}\left(\Gamma_k(T)\sqrt{ T}+\mathbb{E}[\tau]\Gamma_k(T)\right)$ reported in (Verma et al., 2022). In particular, for very non-smooth kernels, the information gain grows almost linearly in time, trivializing the existing results. We also validate our theoretical results with simulations.} }
Endnote
%0 Conference Paper %T Delayed Feedback in Kernel Bandits %A Sattar Vakili %A Danyal Ahmed %A Alberto Bernacchia %A Ciara Pike-Burke %B Proceedings of the 40th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2023 %E Andreas Krause %E Emma Brunskill %E Kyunghyun Cho %E Barbara Engelhardt %E Sivan Sabato %E Jonathan Scarlett %F pmlr-v202-vakili23a %I PMLR %P 34779--34792 %U https://proceedings.mlr.press/v202/vakili23a.html %V 202 %X Black box optimisation of an unknown function from expensive and noisy evaluations is a ubiquitous problem in machine learning, academic research and industrial production. An abstraction of the problem can be formulated as a kernel based bandit problem (also known as Bayesian optimisation), where a learner aims at optimising a kernelized function through sequential noisy observations. The existing work predominantly assumes feedback is immediately available; an assumption which fails in many real world situations, including recommendation systems, clinical trials and hyperparameter tuning. We consider a kernel bandit problem under stochastically delayed feedback, and propose an algorithm with $\tilde{\mathcal{O}}\left(\sqrt{\Gamma_k(T) T}+\mathbb{E}[\tau]\right)$ regret, where $T$ is the number of time steps, $\Gamma_k(T)$ is the maximum information gain of the kernel with $T$ observations, and $\tau$ is the delay random variable. This represents a significant improvement over the state of the art regret bound of $\tilde{\mathcal{O}}\left(\Gamma_k(T)\sqrt{ T}+\mathbb{E}[\tau]\Gamma_k(T)\right)$ reported in (Verma et al., 2022). In particular, for very non-smooth kernels, the information gain grows almost linearly in time, trivializing the existing results. We also validate our theoretical results with simulations.
APA
Vakili, S., Ahmed, D., Bernacchia, A. & Pike-Burke, C.. (2023). Delayed Feedback in Kernel Bandits. Proceedings of the 40th International Conference on Machine Learning, in Proceedings of Machine Learning Research 202:34779-34792 Available from https://proceedings.mlr.press/v202/vakili23a.html.

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