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## Bayesian Optimistic Optimisation with Exponentially Decaying Regret

### Hung Tran-The · Sunil Gupta · Santu Rana · Svetha Venkatesh

Keywords: [ Gaussian Processes and Bayesian non-parametrics ]

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 Mat\'ern 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.

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