Bayesian global optimization (BGO) is an efficient surrogate-assisted
technique for problems involving expensive evaluations. A parallel technique
can be used to parallelly evaluate the true-expensive objective functions in
one iteration to boost the execution time. An effective and straightforward
approach is to design an acquisition function that can evaluate the performance
of a bath of multiple solutions, instead of a single point/solution, in one
iteration. This paper proposes five alternatives of \emph{Probability of
Improvement} (PoI) with multiple points in a batch (q-PoI) for multi-objective
Bayesian global optimization (MOBGO), taking the covariance among multiple
points into account. Both exact computational formulas and the Monte Carlo
approximation algorithms for all proposed q-PoIs are provided. Based on the
distribution of the multiple points relevant to the Pareto-front, the
position-dependent behavior of the five q-PoIs is investigated. Moreover, the
five q-PoIs are compared with the other nine state-of-the-art and recently
proposed batch MOBGO algorithms on twenty bio-objective benchmarks. The
empirical experiments on different variety of benchmarks are conducted to
demonstrate the effectiveness of two greedy q-PoIs ($\kpoi_{\mbox{best}}$ and
$\kpoi_{\mbox{all}}$) on low-dimensional problems and the effectiveness of two
explorative q-PoIs ($\kpoi_{\mbox{one}}$ and $\kpoi_{\mbox{worst}}$) on
high-dimensional problems with difficult-to-approximate Pareto front
boundaries.

本文提出了五种多目标贝叶斯全局优化的概率提高算法 (q-PoI)，适用于多个解点的批量评估，并针对这些算法的位置依赖行为进行了探讨，并通过实证实验证明了两种贪婪型的 q-PoIs 在低维问题上的有效性以及两种探索性的 q-PoIs 在高维问题上的有效性。

使用改进的概率提高选择技术的多目标贝叶斯全局优化并行算法

A Parallel Technique for Multi-objective Bayesian Global Optimization: Using a Batch Selection of Probability of Improvement

In the field of multi-objective optimization algorithms, multi-objective
Bayesian Global Optimization (MOBGO) is an important branch, in addition to
evolutionary multi-objective optimization algorithms (EMOAs). MOBGO utilizes
Gaussian Process models learned from previous objective function evaluations to
decide the next evaluation site by maximizing or minimizing an infill
criterion. A common criterion in MOBGO is the Expected Hypervolume Improvement
(EHVI), which shows a good performance on a wide range of problems, with
respect to exploration and exploitation. However, so far it has been a
challenge to calculate exact EHVI values efficiently. In this paper, an
efficient algorithm for the computation of the exact EHVI for a generic case is
proposed. This efficient algorithm is based on partitioning the integration
volume into a set of axis-parallel slices. Theoretically, the upper bound time
complexities are improved from previously $O (n^2)$ and $O(n^3)$, for two- and
three-objective problems respectively, to $\Theta(n\log n)$, which is
asymptotically optimal. This article generalizes the scheme in higher
dimensional case by utilizing a new hyperbox decomposition technique, which was
proposed by D{\"a}chert et al, EJOR, 2017. It also utilizes a generalization of
the multilayered integration scheme that scales linearly in the number of
hyperboxes of the decomposition. The speed comparison shows that the proposed
algorithm in this paper significantly reduces computation time. Finally, this
decomposition technique is applied in the calculation of the Probability of
Improvement (PoI).

该文章提出了一种基于切片分解技术的高效算法，能够在多目标优化中准确计算期望超体积改进值，达到了渐近最优的时间复杂度，并将此分解技术应用到概率提升计算中。

利用区间分解算法高效计算预期超体积改进值

Efficient Computation of Expected Hypervolume Improvement Using Box  Decomposition Algorithms

Can one parallelize complex exploration exploitation tradeoffs? As an
example, consider the problem of optimal high-throughput experimental design,
where we wish to sequentially design batches of experiments in order to
simultaneously learn a surrogate function mapping stimulus to response and
identify the maximum of the function. We formalize the task as a multi-armed
bandit problem, where the unknown payoff function is sampled from a Gaussian
process (GP), and instead of a single arm, in each round we pull a batch of
several arms in parallel. We develop GP-BUCB, a principled algorithm for
choosing batches, based on the GP-UCB algorithm for sequential GP optimization.
We prove a surprising result; as compared to the sequential approach, the
cumulative regret of the parallel algorithm only increases by a constant factor
independent of the batch size B. Our results provide rigorous theoretical
support for exploiting parallelism in Bayesian global optimization. We
demonstrate the effectiveness of our approach on two real-world applications.

该论文研究了如何在贝叶斯全局优化中利用并行算法解决多臂赌博问题，提出了基于高斯过程的 GP-BUCB 算法，证明了与串行方法相比，该算法的累积遗憾仅增加一个独立于批量的常数因子，并在两个真实世界应用中展示了其有效性。