Pareto Front Learning (PFL) was recently introduced as an effective approach
to obtain a mapping function from a given trade-off vector to a solution on the
Pareto front, which solves the multi-objective optimization (MOO) problem. Due
to the inherent trade-off between conflicting objectives, PFL offers a flexible
approach in many scenarios in which the decision makers can not specify the
preference of one Pareto solution over another, and must switch between them
depending on the situation. However, existing PFL methods ignore the
relationship between the solutions during the optimization process, which
hinders the quality of the obtained front. To overcome this issue, we propose a
novel PFL framework namely PHN-HVI, which employs a hypernetwork to generate
multiple solutions from a set of diverse trade-off preferences and enhance the
quality of the Pareto front by maximizing the Hypervolume indicator defined by
these solutions. The experimental results on several MOO machine learning tasks
show that the proposed framework significantly outperforms the baselines in
producing the trade-off Pareto front.

提出 PHN-HVI 框架，利用超网络从多样化的权衡偏好生成多个解并最大化这些解定义的超体积指标以提高 Pareto 前沿的质量，在多个 MOO 机器学习任务上实验结果表明，与基线方法相比，该框架显著提高了产生权衡 Pareto 前沿的性能。

利用多样本超网络改善 Pareto 前沿学习

Improving Pareto Front Learning via Multi-Sample Hypernetworks

We propose a new approach to the computation of the hypervolume indicator,
based on partitioning the dominated region into a set of axis-parallel
hyperrectangles or boxes. We present a nonincremental algorithm and an
incremental algorithm, which allows insertions of points, whose time
complexities are $O(n^{\lfloor \frac{p-1}{2} \rfloor+1})$ and $O(n^{\lfloor
\frac{p}{2} \rfloor+1})$, respectively. While the theoretical complexity of
such a method is lower bounded by the complexity of the partition, which is, in
the worst-case, larger than the best upper bound on the complexity of the
hypervolume computation, we show that it is practically efficient. In
particular, the nonincremental algorithm competes with the currently most
practically efficient algorithms. Finally, we prove an enhanced upper bound of
$O(n^{p-1})$ and a lower bound of $\Omega (n^{\lfloor \frac{p}{2}\rfloor} \log
n )$ for $p \geq 4$ on the worst-case complexity of the WFG algorithm.

提出了一种利用轴对齐超矩形进行支配区域划分的新方法来计算超体积指标，包括一个非增量算法和一个增量算法。虽然其理论复杂度受到了分区复杂度的下界限制，但该方法在实践中具有高效性。此外，证明了 WFG 算法的最坏情况下的复杂度的改进上限和下限。

计算超体积指标的箱形分解算法

A Box Decomposition Algorithm to Compute the Hypervolume Indicator

We consider the computation of the volume of the union of high-dimensional
geometric objects. While showing that this problem is #P-hard already for very
simple bodies (i.e., axis-parallel boxes), we give a fast FPRAS for all objects
where one can: (1) test whether a given point lies inside the object, (2)
sample a point uniformly, (3) calculate the volume of the object in polynomial
time. All three oracles can be weak, that is, just approximate. This implies
that Klee's measure problem and the hypervolume indicator can be approximated
efficiently even though they are #P-hard and hence cannot be solved exactly in
time polynomial in the number of dimensions unless P=NP. Our algorithm also
allows to approximate efficiently the volume of the union of convex bodies
given by weak membership oracles.
For the analogous problem of the intersection of high-dimensional geometric
objects we prove #P-hardness for boxes and show that there is no multiplicative
polynomial-time $2^{d^{1-\epsilon}}$-approximation for certain boxes unless
NP=BPP, but give a simple additive polynomial-time $\epsilon$-approximation.

本研究探讨高维几何对象的求并集体积问题。本文给出了一个快速的 FPRAS 算法，可以有效地逼近 Klee's measure problem 和 hypervolume indicator，并能够逼近由弱成员资格证明给出的凸体的并集体积。同时，作者还探讨了高维几何对象的交集计算问题，并证明了其 #P 难度。