改进高斯噪声下贝叶斯优化的累积遗憾上界，提升算法收敛速度。
On Improved Regret Bounds In Bayesian Optimization with Gaussian Noise
- 基于频率学派设定，推导高斯过程预测误差的新点态界。
- 证明了GP-UCB与GP-TS在高斯噪声下的累积遗憾上界更优。
- 结果可推广至期望改进等方法，适用于带噪声的优化场景。
带有高斯过程（GP）代理模型的贝叶斯优化（BO）是一种强大的黑箱优化方法。采集函数是BO算法的核心，决定新样本的选择方式。常用的采集函数包括上置信界（UCB）和汤普森采样（TS）。现有研究主要关注在贝叶斯与频率学派设定下目标函数的累积遗憾。本文在频率学派设定下，针对高斯噪声，建立了高斯过程预测误差的新点态上界。据此，我们证明了GP-UCB与GP-TS的累积遗憾上界得到改进。值得注意的是，该预测误差上界可应用于一般贝叶斯优化算法及收敛性分析，例如带噪声时期望改进（EI）的渐近收敛性。
原文摘要 · Abstract (English)
Bayesian optimization (BO) with Gaussian process (GP) surrogate models is a powerful black-box optimization method. Acquisition functions are a critical part of a BO algorithm as they determine how the new samples are selected. Some of the most widely used acquisition functions include upper confidence bound (UCB) and Thompson sampling (TS). The convergence analysis of BO algorithms has focused on the cumulative regret under both the Bayesian and frequentist settings for the objective. In this paper, we establish new pointwise bounds on the prediction error of GP under the frequentist setting with Gaussian noise. Consequently, we prove improved convergence rates of cumulative regret bound for both GP-UCB and GP-TS. Of note, the new prediction error bound under Gaussian noise can be applied to general BO algorithms and convergence analysis, e.g., the asymptotic convergence of expected improvement (EI) with noise.
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