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：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超参数取值，而这些超参数通常预先未知，需从观测数据中估计。然而实际应用中，由于BO采用有偏数据采样策略，这些估计可能产生偏差，导致性能下降并破坏BO的次线性全局收敛保证。针对该问题，我们提出一种新型BO方法——即便真实GP超参数预先未知且需从观测数据估计，该方法仍能以次线性速率收敛至目标函数全局最优解。该方法采用多臂赌博机技术（EXP3）向BO过程添加随机数据点，并设计新型GP超参数估计训练损失函数以确保一致性估计。我们进一步对所提方法进行了理论分析。最后，通过合成问题与真实世界问题的实验验证，我们证明该方法性能优于现有方案。