Bayesian Optimization of Robustness Measures Using Randomized GP-UCB-based Algorithms under Input Uncertainty

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.