Bayesian optimization based on the Gaussian process upper confidence bound (GP-UCB) offers a theoretical guarantee for optimizing black-box functions. In practice, however, black-box functions often involve input uncertainty. To handle such cases, GP-UCB can be extended to optimize evaluation criteria known as robustness measures. However, GP-UCB-based methods for robustness measures require a trade-off parameter, $\beta$, which, as in the original GP-UCB, must be set sufficiently large to ensure theoretical validity. In this study, we propose randomized robustness measure GP-UCB (RRGP-UCB), a novel method that samples $\beta$ from a chi-squared-based probability distribution. This approach eliminates the need to explicitly specify $\beta$. Notably, the expected value of $\beta$ under this distribution is not excessively large. Furthermore, we show that RRGP-UCB provides tight bounds on the expected regret between the optimal and estimated solutions. Numerical experiments demonstrate the effectiveness of the proposed method.

翻译：基于高斯过程上置信界（GP-UCB）的贝叶斯优化为黑箱函数优化提供了理论保证。然而在实际应用中，黑箱函数常涉及输入不确定性。为处理此类情况，GP-UCB可扩展用于优化称为稳健性度量的评估准则。但基于GP-UCB的稳健性度量优化方法需要权衡参数β，该参数与原始GP-UCB相同，必须设定足够大以确保理论有效性。本研究提出随机化稳健性度量GP-UCB（RRGP-UCB）这一新方法，通过从卡方分布中采样β值，无需显式指定该参数。值得注意的是，该分布下β的期望值不会过大。此外，我们证明RRGP-UCB能为最优解与估计解之间的期望遗憾提供紧致界。数值实验验证了所提方法的有效性。