首次给出带噪声的贝叶斯优化期望改进的收敛速率理论
On the convergence rate of noisy Bayesian Optimization with Expected Improvement
- 在高斯过程先验下分析期望改进的收敛性
- 推导出带噪声观测下的渐近误差界及收敛速率
- 揭示非凸改进函数的探索与利用权衡机制
期望改进（EI）是贝叶斯优化中最常用的获取函数之一。尽管其在实际应用中表现优异，但关于其理论收敛行为和收敛速率的重要问题仍悬而未决。本文在三个关键方面推进了对EI收敛性的理论研究：首先，考虑符合高斯过程（GP）先验假设的目标函数，而现有工作多集中于再生核希尔伯特空间（RKHS）中的函数；其次，在GP先验假设下首次建立了带噪声观测的GP-EI的渐近误差界及其收敛速率；第三，通过分析非凸EI函数的探索与利用特性，给出了噪声自由与带噪声情形下GP-EI的改进误差界。
原文摘要 · Abstract (English)
Expected improvement (EI) is one of the most widely used acquisition functions in Bayesian optimization (BO). Despite its proven success in applications for decades, important open questions remain on the theoretical convergence behaviors and rates for EI. In this paper, we contribute to the convergence theory of EI in three novel and critical areas. First, we consider objective functions that fit under the Gaussian process (GP) prior assumption, whereas existing works mostly focus on functions in the reproducing kernel Hilbert space (RKHS). Second, we establish for the first time the asymptotic error bound and its corresponding rate for GP-EI with noisy observations under the GP prior assumption. Third, by investigating the exploration and exploitation properties of the non-convex EI function, we establish improved error bounds of GP-EI for both the noise-free and noisy cases.
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