Converge Then Diversify: Decoupling Convergence and Diversity in Multi-Objective Bayesian Optimisation

先收敛后多样化：多目标贝叶斯优化中收敛性与多样性的解耦

Abstract: Multi-objective Bayesian optimisation (MOBO) is a sample-efficient approach for optimising expensive black-box functions with multiple objectives. In MOBO, the goal is to adequately approximate the Pareto front; that is, to obtain a high-quality solution set with 1) good convergence (closeness to the Pareto front) and 2) good diversity (spread across the Pareto front).

摘要： 多目标贝叶斯优化（MOBO）是一种用于优化昂贵黑盒多目标函数的样本高效方法。在 MOBO 中，目标是充分逼近帕累托前沿（Pareto front）；即获得一个高质量的解集，使其具备：1）良好的收敛性（接近帕累托前沿）和 2）良好的多样性（在帕累托前沿上分布广泛）。

Existing MOBO methods typically aim to accomplish these two tasks simultaneously, i.e., driving the search towards the Pareto front while maintaining a diverse set of nondominated solutions, such that the solutions, ideally, can gradually approach the entire front. When sufficient search budgets are available, this approach is effective. However, considering both convergence and diversity throughout the search is not easy and requires careful design. Under very tight budgets, there may not be enough solutions generated to be able to simultaneously approach the entire Pareto front.

现有的 MOBO 方法通常旨在同时完成这两项任务，即在推动搜索向帕累托前沿靠拢的同时，保持一组多样化的非支配解，从而使解集理想地逐渐逼近整个前沿。当搜索预算充足时，这种方法是有效的。然而，在整个搜索过程中同时兼顾收敛性和多样性并非易事，需要精心的设计。在预算非常紧张的情况下，生成的解可能不足以同时逼近整个帕累托前沿。

To address this issue, this paper proposes a \textit{converge-then-diversify} (CTD) approach that decouples convergence and diversity into two stages. In the first stage, CTD focuses on convergence, aiming to quickly drive the search toward a single point on the Pareto front. In the second stage, CTD focuses on diversity, aiming to spread solutions across the front.

为了解决这个问题，本文提出了一种“先收敛后多样化”（CTD）的方法，将收敛性和多样性解耦为两个阶段。在第一阶段，CTD 专注于收敛，旨在快速将搜索引导至帕累托前沿上的某一点。在第二阶段，CTD 专注于多样性，旨在将解扩展到整个前沿。

We present two simple instantiations of CTD by using widely adopted acquisition functions in the area. Experimental results show that, across all 446 pairwise comparisons, CTD statistically outperforms state-of-the-art methods in 72.9% of the cases, performs equivalently in 21.1%, and is statistically worse in only 6.1%, with the advantage being particularly evident in settings with very tight evaluation budgets or in high-dimensional problems.

我们通过使用该领域广泛采用的采集函数，展示了 CTD 的两个简单实例化。实验结果表明，在全部 446 组两两比较中，CTD 在 72.9% 的情况下在统计学上优于现有最先进的方法，21.1% 的情况下表现相当，仅在 6.1% 的情况下表现较差；这种优势在评估预算非常紧张或高维问题的设置中尤为明显。