Efficient global optimization is a widely used method for optimizing expensive black-box functions. In this paper, we study the worst-case oracle complexity of the efficient global optimization problem. In contrast to existing kernel-specific results, we derive a unified lower bound for the oracle complexity of efficient global optimization in terms of the metric entropy of a ball in its corresponding reproducing kernel Hilbert space. Moreover, we show that this lower bound nearly matches the upper bound attained by non-adaptive search algorithms, for the commonly used squared exponential kernel and the Matérn kernel with a large smoothness parameter \(\nu \). This matching is up to a replacement of d/2 by d and a logarithmic term \(\log \frac{R}{\epsilon }\), where d is the dimension of input space, R is the upper bound for the norm of the unknown black-box function, and \(\epsilon \) is the desired accuracy. That is to say, our lower bound is nearly optimal for these kernels.

高效的全局优化是一种广泛使用的优化昂贵的黑盒函数的方法。在本文中，我们研究了高效全局优化问题的最坏情况预言复杂度。与现有的特定于内核的结果相比，我们根据球在其相应的再生内核希尔伯特空间中的度量熵，得出了有效全局优化的预言复杂性的统一下界。此外，我们表明，对于常用的平方指数核和具有大平滑参数\(\nu \)的 Matérn 核，该下界几乎与非自适应搜索算法获得的上限相匹配。这种匹配取决于将d /2 替换为d和对数项\(\log \frac{R}{\epsilon }\)，其中d是输入空间的维度，R是范数的上限未知黑盒函数的值，\(\epsilon \)是所需的精度。也就是说，我们的下界对于这些内核来说几乎是最佳的。