In this paper, we study the problem of estimating the normalizing constant $\int e^{-\lambda f(x)}dx$ through queries to the black-box function $f$, where $f$ belongs to a reproducing kernel Hilbert space (RKHS), and $\lambda$ is a problem parameter. We show that to estimate the normalizing constant within a small relative error, the level of difficulty depends on the value of $\lambda$: When $\lambda$ approaches zero, the problem is similar to Bayesian quadrature (BQ), while when $\lambda$ approaches infinity, the problem is similar to Bayesian optimization (BO). More generally, the problem varies between BQ and BO. We find that this pattern holds true even when the function evaluations are noisy, bringing new aspects to this topic. Our findings are supported by both algorithm-independent lower bounds and algorithmic upper bounds, as well as simulation studies conducted on a variety of benchmark functions.

翻译：本文研究通过查询黑箱函数 $f$ 来估计归一化常数 $\int e^{-\lambda f(x)}dx$ 的问题，其中 $f$ 属于再生核希尔伯特空间（RKHS），$\lambda$ 为问题参数。我们证明，要在较小的相对误差内估计归一化常数，其难度取决于 $\lambda$ 的取值：当 $\lambda$ 趋近于零时，问题类似于贝叶斯求积（BQ）；而当 $\lambda$ 趋近于无穷时，问题则类似于贝叶斯优化（BO）。更一般地，该问题在BQ与BO之间连续变化。我们发现，即使在函数评价存在噪声的情况下，这一规律依然成立，为该课题带来了新的视角。我们的结论得到了算法无关下界、算法上界以及在多种基准函数上开展的仿真研究的共同支持。