BOtied: Multi-objective Bayesian optimization with tied multivariate ranks (2306.00344v2)

Abstract: Many scientific and industrial applications require the joint optimization of multiple, potentially competing objectives. Multi-objective Bayesian optimization (MOBO) is a sample-efficient framework for identifying Pareto-optimal solutions. At the heart of MOBO is the acquisition function, which determines the next candidate to evaluate by navigating the best compromises among the objectives. In this paper, we show a natural connection between non-dominated solutions and the extreme quantile of the joint cumulative distribution function (CDF). Motivated by this link, we propose the Pareto-compliant CDF indicator and the associated acquisition function, BOtied. BOtied inherits desirable invariance properties of the CDF, and an efficient implementation with copulas allows it to scale to many objectives. Our experiments on a variety of synthetic and real-world problems demonstrate that BOtied outperforms state-of-the-art MOBO acquisition functions while being computationally efficient for many objectives.

• The paper presents a new CDF indicator that robustly evaluates Pareto sets invariant to objective transformations.
• It leverages copula-based multivariate ranks to efficiently navigate high-dimensional objective spaces with reduced computation times.
• Empirical results demonstrate that BOtied outperforms noisy EHVI and ParEGO on both synthetic benchmarks and real-world applications.

BOtied: Multi-objective Bayesian Optimization with Tied Multivariate Ranks

The paper introduces BOtied, an innovative acquisition function for Multi-objective Bayesian Optimization (MOBO) that addresses scaling challenges prevalent in optimizing scenarios featuring multiple objectives. The core innovation is the use of multivariate ranks through copulas to efficiently approximate Pareto-optimal solutions. This introduction of a CDF-based acquisition approach addresses the limitations of existing methodologies like expected hypervolume improvement (EHVI) and entropy search, which fail to scale efficiently with an increasing number of objectives.

Summary of Contributions

1. Introduction of the CDF Indicator: The paper presents a new Pareto-compliant performance criterion, the CDF indicator. This metric evaluates the quality of approximate Pareto sets. Unlike traditional hypervolume indicators, the CDF indicator remains invariant under transformations or rescalings of the objectives, thus offering robustness in diverse settings.
2. Multivariate Rank Acquisition Function: BOtied leverages copulas, allowing it to efficiently navigate high-dimensional spaces. By evaluating the joint CDF using copulas, it maintains computational tractability even when the number of objectives increases. The method draws a natural connection between the highest multivariate rank and the Pareto-optimal solutions.
3. Outperformance and Computational Advantage: The empirical evaluation demonstrates that BOtied outperforms existing acquisition functions such as noisy EHVI (NEHVI) and noisy random scalarization (like ParEGO) in both synthetic and real-data experiments. It achieves a robust performance balance across various benchmarks while being significantly more efficient computationally.

Evaluation and Results

The experimental results showcase BOtied’s capability in both low-dimensional and complex high-dimensional MOBO problems. The paper benchmarks BOtied against established acquisition strategies across multiple datasets, including classical test functions like Branin-Currin and DTLZ, and real-world datasets such as Penicillin and Caco-2+. The robustness of BOtied is evident in settings that involve dominance of multiple competing objectives.

The efficiency of BOtied is further highlighted by significantly reduced computation times compared to methods involving complex objective space decompositions. The ability to remain robust against objective transformations makes BOtied particularly attractive for practical applications where scale or measurement unit changes occur frequently.

Implications and Future Directions

The introduction of BOtied opens up new pathways for tackling high-dimensional optimization problems. Its robustness and computational efficiency suggest several implications:

• Practical Applications: BOtied’s robustness against objective scale variations is crucial for applications in areas like material design and hyperparameter tuning, where objective dimensionality changes dynamically during optimization.
• Dynamic Domains: The ability to incorporate copula-based modeling suggests future extensions into domains with more dynamic and stochastic environments, potentially leading to the development of adaptive acquisition functions that can learn the structure inherent in changing environments.
• Integration in Automated Systems: The use of multivariate ranks offers intuitive integration points for automated systems that must adaptively balance competing design objectives, such as in automated machine learning systems and complex system design.

In future work, further exploration of copula families and vine copula structures could enhance the flexibility and robustness of the BOtied methodology, as well as the integration of multivariate ranks with existing multi-fidelity modeling frameworks. Such developments have the potential to extend the scope and application of BOtied in real-world decision-making systems. Additionally, exploring connections between copulas and other probabilistic estimation techniques in machine learning could provide new insights into performance improvements in MOBO, further cementing BOtied's applicability.