Bayesian Optimistic Optimisation with Exponentially Decaying Regret

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 O(fraclogN√N) to mathcal O(e^-√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^-√N) under the assumption that the objective function is sampled from a Gaussian process with a Mat'ern kernel with smoothness parameter nu > 4 +fracD2, 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.