Single-objective black box optimization (also known as zeroth-order optimization) is the process of minimizing a scalar objective $f(x)$, given evaluations at adaptively chosen inputs $x$. In this paper, we consider multi-objective optimization, where $f(x)$ outputs a vector of possibly competing objectives and the goal is to converge to the Pareto frontier. Quantitatively, we wish to maximize the standard hypervolume indicator metric, which measures the dominated hypervolume of the entire set of chosen inputs. In this paper, we introduce a novel scalarization function, which we term the hypervolume scalarization, and show that drawing random scalarizations from an appropriately chosen distribution can be used to efficiently approximate the hypervolume indicator metric. We utilize this connection to show that Bayesian optimization with our scalarization via common acquisition functions, such as Thompson Sampling or Upper Confidence Bound, provably converges to the whole Pareto frontier by deriving tight hypervolume regret bounds on the order of $\widetilde{O}(\sqrt{T})$. Furthermore, we highlight the general utility of our scalarization framework by showing that any provably convergent single-objective optimization process can be effortlessly converted to a multi-objective optimization process with provable convergence guarantees.

本文提出了一种称为hypervolume scalarization的标量化函数，并显示出如何从适当选择的分布中绘制随机标量化以有效地近似hypervolume indicator metric，进而用此联系说明Bayesian optimization具有证明收敛到整个Pareto frontier的保证收敛性，同时凸显了标量化框架的一般实用性。

随机超体积标量化技术的可证明多目标黑盒优化