We consider the problem of optimizing a grey-box objective function, i.e., nested function composed of both black-box and white-box functions. A general formulation for such grey-box problems is given, which covers the existing grey-box optimization formulations as special cases. We then design an optimism-driven algorithm to solve it. Under certain regularity assumptions, our algorithm achieves similar regret bound as that for the standard black-box Bayesian optimization algorithm, up to a constant multiplicative term depending on the Lipschitz constants of the functions considered. We further extend our method to the constrained case and discuss special cases. For the commonly used kernel functions, the regret bounds allow us to derive a convergence rate to the optimal solution. Experimental results show that our grey-box optimization method empirically improves the speed of finding the global optimal solution significantly, as compared to the standard black-box optimization algorithm.

翻译：我们考虑优化灰箱目标函数的问题，即由黑箱函数和白箱函数嵌套组合而成的函数。本文给出此类灰箱问题的一般性表述，该表述涵盖现有灰箱优化公式作为特例。进而设计了一种基于乐观驱动的算法求解该问题。在特定正则性假设下，我们的算法与标准黑箱贝叶斯优化算法相比，其遗憾界仅相差一个取决于所考虑函数利普希茨常数的常数倍乘项。进一步将方法扩展至约束情形并讨论特例。对于常用核函数，该遗憾界允许推导出收敛至最优解的收敛速率。实验结果表明，与标准黑箱优化算法相比，所提出的灰箱优化方法在经验上显著提升全局最优解的搜索速度。