Bayesian optimization based on Gaussian process upper confidence bound (GP-UCB) has a theoretical guarantee for optimizing black-box functions. Black-box functions often have input uncertainty, but even in this case, GP-UCB can be extended to optimize evaluation measures called robustness measures. However, GP-UCB-based methods for robustness measures include a trade-off parameter $\beta$, which must be excessively large to achieve theoretical validity, just like the original GP-UCB. In this study, we propose a new method called randomized robustness measure GP-UCB (RRGP-UCB), which samples the trade-off parameter $\beta$ from a probability distribution based on a chi-squared distribution and avoids explicitly specifying $\beta$. The expected value of $\beta$ is not excessively large. Furthermore, we show that RRGP-UCB provides tight bounds on the expected value of regret based on the optimal solution and estimated solutions. Finally, we demonstrate the usefulness of the proposed method through numerical experiments.

翻译：基于高斯过程上置信界（GP-UCB）的贝叶斯优化对于黑箱函数优化具有理论保证。黑箱函数常存在输入不确定性，但即使在此情况下，GP-UCB仍可扩展用于优化称为鲁棒性度量的评估指标。然而，基于GP-UCB的鲁棒性度量优化方法包含一个权衡参数$\beta$——正如原始GP-UCB那样，该参数必须取极大值才能保证理论有效性。本研究提出一种名为随机化鲁棒性度量GP-UCB（RRGP-UCB）的新方法，该方法基于卡方分布从概率分布中采样权衡参数$\beta$，从而避免显式设定$\beta$值。$\beta$的期望值不会过度增大。此外，我们证明RRGP-UCB能够基于最优解与估计解给出紧致的期望遗憾界。最后通过数值实验验证了所提方法的有效性。