Bayesian Optimistic Optimisation with Exponentially Decaying Regret
Hung Tran-The, Sunil Gupta, Santu Rana, Svetha Venkatesh
Abstract
Bayesian optimisation (BO) is a well known algorithm for finding the global optimum of expensive, black-box functions. The current practical BO algorithms have regret bounds ranging from $\mathcal{O}(\frac{logN}{\sqrt{N}})$ to $\mathcal O(e^{-\sqrt{N}})$, where $N$ is the number of evaluations. This paper explores the possibility of improving the regret bound in the noise-free setting by intertwining concepts from BO and optimistic optimisation methods which are based on partitioning the search space. We propose the BOO algorithm, a first practical approach which can achieve an exponential regret bound with order $\mathcal O(N^{-\sqrt{N}})$ under the assumption that the objective function is sampled from a Gaussian process with a Matérn kernel with smoothness parameter $\nu > 4 +\frac{D}{2}$, where $D$ is the number of dimensions. We perform experiments on optimisation of various synthetic functions and machine learning hyperparameter tuning tasks and show that our algorithm outperforms baselines.
BibTeX
@InProceedings{pmlr-v139-tran-the21a,
title = {Bayesian Optimistic Optimisation with Exponentially Decaying Regret},
author = {Tran-The, Hung and Gupta, Sunil and Rana, Santu and Venkatesh, Svetha},
booktitle = {Proceedings of the 38th International Conference on Machine Learning},
pages = {10390--10400},
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/tran-the21a/tran-the21a.pdf},
url = {https://proceedings.mlr.press/v139/tran-the21a.html},
abstract = {Bayesian optimisation (BO) is a well known algorithm for finding the global optimum of expensive, black-box functions. The current practical BO algorithms have regret bounds ranging from $\mathcal{O}(\frac{logN}{\sqrt{N}})$ to $\mathcal O(e^{-\sqrt{N}})$, where $N$ is the number of evaluations. This paper explores the possibility of improving the regret bound in the noise-free setting by intertwining concepts from BO and optimistic optimisation methods which are based on partitioning the search space. We propose the BOO algorithm, a first practical approach which can achieve an exponential regret bound with order $\mathcal O(N^{-\sqrt{N}})$ under the assumption that the objective function is sampled from a Gaussian process with a Matérn kernel with smoothness parameter $\nu > 4 +\frac{D}{2}$, where $D$ is the number of dimensions. We perform experiments on optimisation of various synthetic functions and machine learning hyperparameter tuning tasks and show that our algorithm outperforms baselines.}
}