Bayesian optimization is a principled optimization strategy for a black-box
objective function. It shows its effectiveness in a wide variety of real-world
applications such as scientific discovery and experimental design. In general,
the performance of Bayesian optimization is assessed by regret-based metrics
such as instantaneous, simple, and cumulative regrets. These metrics only rely
on function evaluations, so that they do not consider geometric relationships
between query points and global solutions, or query points themselves. Notably,
they cannot discriminate if multiple global solutions are successfully found.
Moreover, they do not evaluate Bayesian optimization's abilities to exploit and
explore a search space given. To tackle these issues, we propose four new
geometric metrics, i.e., precision, recall, average degree, and average
distance. These metrics allow us to compare Bayesian optimization algorithms
considering the geometry of both query points and global optima, or query
points. However, they are accompanied by an extra parameter, which needs to be
carefully determined. We therefore devise the parameter-free forms of the
respective metrics by integrating out the additional parameter. Finally, we
empirically validate that our proposed metrics can provide more convincing
interpretation and understanding of Bayesian optimization algorithms from
distinct perspectives, compared to the conventional metrics.

贝叶斯优化算法的效果评估依赖于遗憾度量，然而这些度量无法考虑查询点与全局解的几何关系，以及贝叶斯优化算法对搜索空间的开发和探索能力。为解决这些问题，本研究提出了四个新的几何度量，并通过集成额外参数来得到无需参数的指标形式。经实验证明，这些新的度量方法能更全面、有说服力地解释和理解贝叶斯优化算法。