Gaussian process (GP) based Bayesian optimization (BO) is a powerful method for optimizing black-box functions efficiently. The practical performance and theoretical guarantees associated with 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 commonly 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 global optimum of the objective function 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 unbiased estimation from the observed data. 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：高斯过程（GP）驱动的贝叶斯优化（BO）是一种高效优化黑箱函数的强有力方法。该方法的实际性能与理论保证依赖于正确的GP超参数取值，而这些超参数通常事先未知，需要从观测数据中估计得到。然而在实际应用中，由于BO常用的有偏数据采样策略，这些估计可能存在偏差，从而导致性能退化，并破坏BO的次线性全局收敛保证。针对这一问题，我们提出了一种新型BO方法，即使在真实GP超参数未知且需从观测数据估计的情况下，仍能以次线性速率收敛至目标函数的全局最优解。该方法采用多臂老虎机技术（EXP3）向BO过程中添加随机数据点，并设计了一种新颖的GP超参数估计训练损失函数，确保从观测数据中获得无偏估计。我们进一步对所提方法进行了理论分析。最后，通过合成与实际问题的实验验证表明，本方法在性能上优于现有各类方案。