Bayesian Optimization (BO) has proven to be very successful at optimizing a static, noisy, costly-to-evaluate black-box function $f : \mathcal{S} \to \mathbb{R}$. However, optimizing a black-box which is also a function of time (i.e., a dynamic function) $f : \mathcal{S} \times \mathcal{T} \to \mathbb{R}$ remains a challenge, since a dynamic Bayesian Optimization (DBO) algorithm has to keep track of the optimum over time. This changes the nature of the optimization problem in at least three aspects: (i) querying an arbitrary point in $\mathcal{S} \times \mathcal{T}$ is impossible, (ii) past observations become less and less relevant for keeping track of the optimum as time goes by and (iii) the DBO algorithm must have a high sampling frequency so it can collect enough relevant observations to keep track of the optimum through time. In this paper, we design a Wasserstein distance-based criterion able to quantify the relevancy of an observation with respect to future predictions. Then, we leverage this criterion to build W-DBO, a DBO algorithm able to remove irrelevant observations from its dataset on the fly, thus maintaining simultaneously a good predictive performance and a high sampling frequency, even in continuous-time optimization tasks with unknown horizon. Numerical experiments establish the superiority of W-DBO, which outperforms state-of-the-art methods by a comfortable margin.

翻译：贝叶斯优化（BO）已被证明在优化静态、含噪声、评估代价高昂的黑盒函数 $f : \mathcal{S} \to \mathbb{R}$ 方面非常成功。然而，优化一个同时也是时间函数的黑盒（即动态函数）$f : \mathcal{S} \times \mathcal{T} \to \mathbb{R}$ 仍然是一个挑战，因为动态贝叶斯优化（DBO）算法必须随时间跟踪最优解。这至少在三个方面改变了优化问题的性质：(i) 查询 $\mathcal{S} \times \mathcal{T}$ 中的任意点是不可能的，(ii) 随着时间的推移，过去的观测对于跟踪最优解的相关性越来越低，以及 (iii) DBO 算法必须具有较高的采样频率，以便收集足够多的相关观测来随时间跟踪最优解。在本文中，我们设计了一种基于 Wasserstein 距离的准则，能够量化观测对未来预测的相关性。然后，我们利用该准则构建了 W-DBO，这是一种能够实时从数据集中移除无关观测的 DBO 算法，从而即使在未知时间范围的连续时间优化任务中，也能同时保持良好的预测性能和高采样频率。数值实验确立了 W-DBO 的优越性，其性能以显著优势超越了现有最先进方法。