Gaussian process (GP) based Bayesian optimization (BO) is a powerful method for optimizing black-box functions efficiently. The practical performance and theoretical guarantees of this approach depend on having the correct GP hyperparameter values, which are usually unknown in advance and need to be estimated from the observed data. However, in practice, these estimations could be incorrect due to biased data sampling strategies used in BO. This can lead to degraded performance and break the sub-linear global convergence guarantee of BO. To address this issue, we propose a new BO method that can sub-linearly converge to the objective function's global optimum even when the true GP hyperparameters are unknown in advance and need to be estimated from the observed data. Our method uses a multi-armed bandit technique (EXP3) to add random data points to the BO process, and employs a novel training loss function for the GP hyperparameter estimation process that ensures consistent estimation. We further provide theoretical analysis of our proposed method. Finally, we demonstrate empirically that our method outperforms existing approaches on various synthetic and real-world problems.

翻译：基于高斯过程（GP）的贝叶斯优化（BO）是一种高效优化黑箱函数的强大方法。该方法的实际性能与理论保证依赖于正确的GP超参数值，而这些值通常事先未知，需要从观测数据中估计。然而，在实践中，由于BO中使用的数据采样策略可能存在偏差，这些估计可能不正确。这可能导致性能下降，并破坏BO的次线性全局收敛保证。为解决此问题，我们提出了一种新的BO方法，即使在真实GP超参数事先未知且需从观测数据中估计的情况下，该方法仍能以次线性速度收敛至目标函数的全局最优解。我们的方法使用多臂老虎机技术（EXP3）向BO过程中添加随机数据点，并为GP超参数估计过程采用了一种新颖的训练损失函数，以确保估计的一致性。我们进一步提供了所提方法的理论分析。最后，我们通过实验证明，我们的方法在各种合成与真实世界问题上均优于现有方法。