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，也称为基于核的博彩优化。我们研究利用从分布中抽取的随机样本进行域探索的方法。研究表明，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，该结果可能具有独立研究价值。我们进一步开发了一种基于随机探索与域收缩的算法，并在无噪声和有噪声两种设定下建立了其阶最优遗憾保证。在无噪声设定中，我们的分析弥补了现有遗憾性能的差距，从而解决了COLT开放问题。由于随机探索避免了每轮迭代中对非凸采集函数进行昂贵优化来选择查询点，该算法在计算效率上相较于主流方法具有显著优势。