We consider Bayesian optimization using Gaussian Process models, also referred to as kernel-based bandit optimization. We study the methodology of exploring the domain using random samples drawn from a distribution. We show that this random exploration approach achieves the optimal error rates. Our analysis is based on novel concentration bounds in an infinite dimensional Hilbert space established in this work, which may be of independent interest. We further develop an algorithm based on random exploration with domain shrinking and establish its order-optimal regret guarantees under both noise-free and noisy settings. In the noise-free setting, our analysis closes the existing gap in regret performance and thereby resolves a COLT open problem. The proposed algorithm also enjoys a computational advantage over prevailing methods due to the random exploration that obviates the expensive optimization of a non-convex acquisition function for choosing the query points at each iteration.

翻译：我们考虑使用高斯过程模型的贝叶斯优化，也称为基于核的赌博机优化。我们研究了利用从分布中随机抽取的样本进行域探索的方法论。我们证明，这种随机探索方法能够达到最优误差率。我们的分析基于本文建立的无限维希尔伯特空间中的新型浓度界，该成果可能具有独立的研究价值。我们进一步开发了一种结合域收缩的随机探索算法，并在无噪声和有噪声两种场景下建立了其阶最优遗憾保证。在无噪声场景中，我们的分析填补了现有遗憾性能的空白，从而解决了COLT开放问题。与主流方法相比，该算法还享有计算优势，因其随机探索省去了每次迭代中选择查询点时对非凸采集函数进行昂贵优化的过程。