Bayesian optimisation requires fitting a Gaussian process model, which in turn requires specifying prior on the unknown black-box function -- most of the theoretical literature assumes this prior is known. However, it is common to have more than one possible prior for a given black-box function, for example suggested by domain experts with differing opinions. In some cases, the type-II maximum likelihood estimator for selecting prior enjoys the consistency guarantee, but it does not universally apply to all types of priors. If the problem is stationary, one could rely on the Regret Balancing scheme to conduct the optimisation, but in the case of time-varying problems, such a scheme cannot be used. To address this gap in existing research, we propose a novel algorithm, PE-GP-UCB, which is capable of solving time-varying Bayesian optimisation problems even without the exact knowledge of the function's prior. The algorithm relies on the fact that either the observed function values are consistent with some of the priors, in which case it is easy to reject the wrong priors, or the observations are consistent with all candidate priors, in which case it does not matter which prior our model relies on. We provide a regret bound on the proposed algorithm. Finally, we empirically evaluate our algorithm on toy and real-world time-varying problems and show that it outperforms the maximum likelihood estimator, fully Bayesian treatment of unknown prior and Regret Balancing.

翻译：贝叶斯优化需要拟合高斯过程模型，而该模型又需要对未知黑箱函数指定先验——现有理论文献大多假设该先验已知。然而，对于给定黑箱函数存在多个可能先验的情况十分常见，例如不同领域专家可能提出相异的先验建议。在某些情况下，用于选择先验的II型最大似然估计器具有一致性保证，但该性质并不普遍适用于所有先验类型。若问题具有平稳性，可依赖遗憾平衡方案进行优化，但对于时变问题，此类方案则无法适用。为填补现有研究空白，本文提出一种新型算法PE-GP-UCB，该算法即使在没有函数先验精确知识的情况下，仍能求解时变贝叶斯优化问题。该算法的理论基础在于：观测到的函数值要么与部分先验保持一致（此时容易排除错误先验），要么与所有候选先验均保持一致（此时模型依赖何种先验并无影响）。我们为所提算法提供了遗憾界理论证明。最后，通过在仿真和真实世界时变问题上的实证评估，证明本算法在性能上优于最大似然估计器、完全贝叶斯处理的未知先验方法以及遗憾平衡方案。