Science and Engineering applications are typically associated with expensive optimization problems to identify optimal design solutions and states of the system of interest. Bayesian optimization and active learning compute surrogate models through efficient adaptive sampling schemes to assist and accelerate this search task toward a given optimization goal. Both those methodologies are driven by specific infill/learning criteria which quantify the utility with respect to the set goal of evaluating the objective function for unknown combinations of optimization variables. While the two fields have seen an exponential growth in popularity in the past decades, their dualism and synergy have received relatively little attention to date. This paper discusses and formalizes the synergy between Bayesian optimization and active learning as symbiotic adaptive sampling methodologies driven by common principles. In particular, we demonstrate this unified perspective through the formalization of the analogy between the Bayesian infill criteria and active learning criteria as driving principles of both the goal-driven procedures. To support our original perspective, we propose a general classification of adaptive sampling techniques to highlight similarities and differences between the vast families of adaptive sampling, active learning, and Bayesian optimization. Accordingly, the synergy is demonstrated mapping the Bayesian infill criteria with the active learning criteria, and is formalized for searches informed by both a single information source and multiple levels of fidelity. In addition, we provide guidelines to apply those learning criteria investigating the performance of different Bayesian schemes for a variety of benchmark problems to highlight benefits and limitations over mathematical properties that characterize real-world applications.

翻译：科学与工程应用通常涉及昂贵的优化问题，以识别系统感兴趣的最优设计方案与状态。贝叶斯优化与主动学习通过高效的自适应采样方案计算代理模型，以辅助并加速针对给定优化目标的搜索任务。这两种方法均由特定的填充/学习准则驱动，这些准则通过量化评估优化变量未知组合的目标函数相对于设定目标的效用。尽管这两个领域在过去数十年间经历了指数级增长，但它们的二元性及协同作用迄今仍未获得足够关注。本文讨论并形式化了贝叶斯优化与主动学习作为由共同原理驱动的共生自适应采样方法之间的协同关系。特别地，我们通过形式化贝叶斯填充准则与主动学习准则作为两种目标驱动过程指导原则的类比，来论证这一统一视角。为支撑这一原创观点，我们提出了自适应采样技术的通用分类框架，以凸显自适应采样、主动学习与贝叶斯优化三大方法族之间的共性与差异。基于此，我们通过将贝叶斯填充准则映射至主动学习准则来论证其协同性，并针对单信息源及多保真度层级的搜索场景进行了形式化表述。此外，我们提供了应用这些学习准则的指导原则，通过研究不同贝叶斯方案在多种基准问题上的性能表现，以揭示其在表征实际应用场景的数学特性方面的优势与局限。