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.

翻译：Bayesian 最佳化( BO) 是一个众所周知的高效算法, 用于寻找全球最优化的昂贵黑盒功能。 当前实用的 BO 算法具有从$mathcal{O}( frac{logNunsqrt{N ⁇ ) 美元到$mathcal O( e ⁇ -\\ sqrt{N ⁇ ) 美元( $) 的遗憾界限, 也就是评估的数量。 本文探讨是否有可能在无噪音环境下改善遗憾约束, 由来自 BO 和基于树的乐观优化概念相互交错, 后者以分隔搜索空间为基础。 我们提出了 BOO 算法, 这是一种首个实用方法, 可以用 $\ mathcal O( N ⁇ _\\\\\\\\\\\\ sqrt{N ⁇ ) 美元实现指数式的遗憾, 假设目标函数是从高斯进程抽样, 带有光滑度参数的 $\nu > 4 ⁇ c{ D ⁇ 2} 美元, 来改进无声波质参数。 我们对各种合成功能和机器进行测试, 测试。