Bayesian optimisation (BO) is a well-known efficient 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 noiseless setting by intertwining concepts from BO and tree-based optimistic optimisation 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.
Categories: cs.LG stat.ML