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。