Bayesian optimization (BO) has shown impressive results in a variety of applications within low-to-moderate dimensional Euclidean spaces. However, extending BO to high-dimensional settings remains a significant challenge. We address this challenge by proposing a two-step optimization framework. Initially, we identify the effective dimension reduction (EDR) subspace for the objective function using the minimum average variance estimation (MAVE) method. Subsequently, we construct a Gaussian process model within this EDR subspace and optimize it using the expected improvement criterion. Our algorithm offers the flexibility to operate these steps either concurrently or in sequence. In the sequential approach, we meticulously balance the exploration-exploitation trade-off by distributing the sampling budget between subspace estimation and function optimization, and the convergence rate of our algorithm in high-dimensional contexts has been established. Numerical experiments validate the efficacy of our method in challenging scenarios.

翻译：贝叶斯优化在中低维欧氏空间的各种应用中已展现出显著成效，但将其扩展至高维场景仍是一项重大挑战。我们提出双阶段优化框架以应对该难题：首先采用最小平均方差估计（MAVE）方法识别目标函数的有效降维（EDR）子空间；随后在该EDR子空间内构建高斯过程模型，并基于期望改进准则进行优化。本算法允许上述阶段以并行或串行方式灵活执行。在串行方案中，我们通过将采样预算在子空间估计与函数优化之间进行合理分配，精细平衡探索-利用权衡，并建立了算法在高维环境下的收敛速率。数值实验验证了该方法在复杂场景中的有效性。