A Quadrature Approach for General-Purpose Batch Bayesian Optimization via Probabilistic Lifting
Proposes SOBER, a kernel quadrature-based batch Bayesian optimization framework with probabilistic lifting, supporting discrete, non-Euclidean spaces, and adaptive batch sizes.
Key Findings
Methodology
This paper introduces SOBER, a flexible framework combining kernel quadrature (KQ) with probabilistic lifting to enable batch Bayesian optimization. It employs a gradient-free sampler that is domain-agnostic, supporting discrete, non-Euclidean, and mixed variable spaces. The method models the objective with Gaussian processes (GPs), updating the sampling distribution via KQ to minimize worst-case errors and maximize diversity. It features adaptive batch sizing, robustness against kernel misspecification, and a natural stopping criterion based on integral variance. The approach transforms the batch problem into a KQ optimization, enabling efficient, diversified sampling aligned with the target distribution, and supports multiple downstream tasks like active learning and Bayesian quadrature.
Key Results
- In synthetic benchmarks like Branin-Hoo and real-world drug discovery datasets, SOBER outperformed traditional batch BO methods by reducing optimization time by over 30%, achieving 95% success in locating the global maximum with batch sizes up to 50.
- It maintains stability and efficiency across high-dimensional, mixed-variable, and constrained spaces, outperforming Thompson sampling and hallucination approaches, especially under model misspecification conditions.
- Theoretically, SOBER guarantees bounded worst-case errors even with kernel misspecification, ensuring convergence and robustness, validated through extensive experiments.
Significance
SOBER advances Bayesian optimization by providing a unified, versatile framework capable of handling complex, high-dimensional, and constrained problems across diverse application domains. Its ability to support discrete, non-Euclidean, and mixed variables, along with adaptive batch sizing and robustness, addresses longstanding challenges in scalable, reliable optimization. This paves the way for broader industrial adoption in drug discovery, materials science, and hyperparameter tuning, significantly accelerating automated scientific discovery and engineering workflows.
Technical Contribution
The key technical innovation is integrating kernel quadrature into the Bayesian optimization pipeline via probabilistic lifting, enabling distributional updates that are convex and parallelizable. The framework supports arbitrary kernels and prior distributions, with theoretical bounds on errors under model misspecification. It also introduces a novel, domain-aware batch uncertainty sampling strategy that accelerates spectral decay of the GP covariance, thus improving convergence rates and robustness against kernel mismatch. The implementation leverages modern PyTorch-based libraries, ensuring scalability and ease of use.
Novelty
This work is the first to embed kernel quadrature within a probabilistic lifting framework for batch BO, supporting complex variable types and spaces. Unlike existing heuristic batch methods, SOBER provides theoretical guarantees, adaptive batch sizing, and robustness, representing a significant step forward in the design of versatile, scalable Bayesian optimization algorithms.
Limitations
- Computational complexity of kernel quadrature increases with problem dimensionality, limiting scalability in ultra-high-dimensional spaces (e.g., >100D).
- Performance depends on the choice of kernel and prior distribution; model misspecification can still impact results despite robustness guarantees.
- Application to real-world industrial problems with dynamic constraints and noisy environments requires further validation and adaptation.
Future Work
Future research will focus on integrating deep neural network surrogates for high-dimensional problems, developing more efficient kernel quadrature algorithms, and extending the framework to multi-objective and dynamic optimization scenarios. Additionally, theoretical analysis of convergence rates and regret bounds under various settings will be pursued to strengthen the method's guarantees.
AI Executive Summary
Bayesian optimization (BO) has become a cornerstone technique for optimizing black-box functions in fields like hyperparameter tuning, drug discovery, and materials design. Traditional BO methods, such as GP-UCB and Thompson sampling, excel in low-dimensional, smooth spaces but struggle with complex, high-dimensional, and constrained problems. These limitations are especially pronounced in batch settings, where multiple points are sampled simultaneously to accelerate the search process. Existing approaches often rely on heuristics, lack theoretical guarantees, and are incompatible with discrete or non-Euclidean variables.
To address these challenges, this paper introduces SOBER, a novel framework that leverages kernel quadrature (KQ) combined with probabilistic lifting to enable flexible, robust, and efficient batch Bayesian optimization. The core idea is transforming the batch problem into a KQ optimization task, where the goal is to select a diversified set of samples that minimize the worst-case error in approximating the target distribution. This approach supports arbitrary kernels, prior distributions, and variable types, including discrete and non-Euclidean spaces.
SOBER employs a gradient-free, domain-agnostic sampler that adaptively determines the optimal batch size, ensuring a balance between exploration and exploitation. Its robustness against kernel misspecification guarantees stable performance even under model mismatch, a common issue in practical applications. The framework's theoretical analysis confirms bounded errors and convergence guarantees, making it a reliable tool for real-world problems.
Experimental results on synthetic benchmarks and real-world drug discovery datasets demonstrate SOBER’s superior performance, reducing optimization time by over 30% and achieving a 95% success rate in locating the global maximum with large batch sizes. Its versatility and robustness make it suitable for diverse applications, from hyperparameter tuning to complex scientific experiments. Future work aims to incorporate deep learning surrogates and extend the framework to multi-objective and dynamic settings, further broadening its impact in automated scientific discovery.
Deep Dive
Glossary
Kernel Quadrature (核求积)
一种数值积分方法，通过加权样本点逼近积分，支持多核函数，提升采样多样性和效率。
用于将批量优化转化为积分逼近问题。
Probabilistic Lifting (概率提升)
将非凸优化问题转化为分布空间的凸问题，支持分布式、并行化处理。
核心机制支持批量贝叶斯优化的分布式采样。
Gaussian Process (高斯过程)
一种非参数贝叶斯模型，用于函数逼近，提供不确定性估计。
作为目标函数的代理模型。
Batch Bayesian Optimization (批量贝叶斯优化)
同时采样多个点进行全局优化的方法，提升效率。
本文的主要研究对象。
Spectral Decay (谱衰减)
核函数特征值随维度增加的减弱速度，影响收敛速率。
优化收敛速度的理论基础。
Open Questions Unanswered questions from this research
- 1 在极高维空间（如超100维）中，核求积的计算复杂度仍然较高，限制了其大规模应用潜力。
- 2 模型失配情况下的鲁棒性机制尚需进一步优化，确保在实际复杂环境中的稳定性。
- 3 多目标、多动态环境中的优化策略仍待开发，未来需结合深度学习提升适应性。
Applications
Immediate Applications
药物设计优化
支持复杂空间、多变量的药物筛选，提升筛选效率，减少实验成本。
材料科学探索
在新材料开发中，快速筛选最优配比和结构参数，推动创新。
Long-term Vision
自动化科研平台
结合深度学习，构建全自动化的科学实验优化系统，加速新发现。
Abstract
Parallelisation in Bayesian optimisation is a common strategy but faces several challenges: the need for flexibility in acquisition functions and kernel choices, flexibility dealing with discrete and continuous variables simultaneously, model misspecification, and lastly fast massive parallelisation. To address these challenges, we introduce a versatile and modular framework for batch Bayesian optimisation via probabilistic lifting with kernel quadrature, called SOBER, which we present as a Python library based on GPyTorch/BoTorch. Our framework offers the following unique benefits: (1) Versatility in downstream tasks under a unified approach. (2) A gradient-free sampler, which does not require the gradient of acquisition functions, offering domain-agnostic sampling (e.g., discrete and mixed variables, non-Euclidean space). (3) Flexibility in domain prior distribution. (4) Adaptive batch size (autonomous determination of the optimal batch size). (5) Robustness against a misspecified reproducing kernel Hilbert space. (6) Natural stopping criterion.