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 several 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：我们考虑优化灰盒目标函数的问题，即由黑盒函数和白盒函数共同构成的嵌套函数。本文给出此类灰盒问题的通用数学表述，该表述将现有灰盒优化形式作为特例涵盖其中。我们设计了基于乐观原则的算法进行求解。在特定正则性假设下，该算法取得的遗憾界与标准黑盒贝叶斯优化算法相比，仅相差一个取决于所考虑函数Lipschitz常数的常数乘积项。进一步我们将方法扩展至带约束情形，并讨论了若干特例。对于常用核函数，该遗憾界使我们能够推导出对最优解的收敛速率。实验结果表明，与标准黑盒优化算法相比，本文提出的灰盒优化方法在实际应用中能显著提升全局最优解的搜索速度。