Sample-efficient Bayesian Optimisation Using Known Invariances
Incorporating known group invariances into Gaussian process kernels enhances Bayesian optimization sample efficiency, with theoretical bounds and practical validation.
Key Findings
Methodology
This work embeds group invariance directly into Gaussian process kernels by constructing invariant kernels under group actions. It leverages spectral properties of these kernels to derive bounds on the maximum information gain, which quantifies sample efficiency improvements. The approach employs invariant kernels within BO algorithms like MVR and UCB, utilizing Matérn kernels for flexibility across different smoothness levels. Theoretical analysis includes spectral decay assumptions and group symmetry properties, leading to explicit bounds on sample complexity as a function of group size. Empirical validation on synthetic functions with permutation and cyclic invariances, as well as a real-world nuclear fusion system design, demonstrates significant reductions in sample requirements and convergence speed.
Key Results
- Synthetic experiments show that invariant kernels reduce the number of samples needed to reach near-optimal solutions by 2-3 times compared to standard kernels, especially in high-dimensional (6D) tasks. Theoretical bounds on information gain and sample complexity align with empirical results, confirming the linear decrease of sample complexity with group size.
- In nuclear fusion current drive system optimization, the invariant kernel approach outperformed traditional methods, achieving high-performance solutions with over 50% fewer samples. The method successfully handled partial invariance scenarios, maintaining robustness when the full symmetry was not perfectly known.
- Analysis indicates that partial invariance still yields substantial gains, and the bounds derived are tight enough to guide practical algorithm design. The approach scales well with group size, making it suitable for complex symmetry groups in real applications.
Significance
This research advances the theoretical understanding of sample complexity bounds in Bayesian optimization under invariance, providing a systematic way to incorporate known symmetries into kernel design. It bridges the gap between invariant kernel theory and practical optimization, enabling more efficient exploration of high-cost functions. The results have broad implications for scientific fields like nuclear fusion, materials science, and robotics, where leveraging physical symmetries can drastically reduce experimental costs. By establishing bounds on information gain and sample complexity, the work offers a rigorous foundation for future invariant optimization methods, fostering more scalable and robust algorithms.
Technical Contribution
The core technical contribution is the construction of totally invariant kernels based on group actions, combined with spectral analysis to derive bounds on the maximum information gain. The paper extends existing invariant kernel theory to the BO setting, providing explicit bounds on sample complexity for classes of Matérn kernels on manifolds. It introduces a novel framework for analyzing the spectral properties of group-invariant kernels, linking group size to sample efficiency. Theoretical results include upper bounds on the information gain and sample complexity, as well as matching lower bounds, establishing near-optimality. The methodology seamlessly integrates spectral theory, information theory, and kernel design, offering a comprehensive toolkit for invariant BO.
Novelty
This is the first systematic derivation of sample complexity bounds for Bayesian optimization using invariant kernels under general finite group actions. Unlike prior work limited to specific groups or additive kernels, this study provides a broad theoretical framework applicable to a wide class of symmetry groups and kernels. It introduces a new spectral analysis approach that relates the group size to information gain, offering explicit bounds that scale linearly with the inverse of the group order. The combination of rigorous theoretical bounds with practical algorithms tailored for invariance represents a significant step forward in the field, bridging the gap between invariant kernel theory and scalable BO.
Limitations
- The approach assumes the invariance is perfectly known and the kernel is exactly invariant, which may not hold in real-world scenarios with model misspecification or approximate symmetries.
- Computational complexity grows with the size of the group, especially for large symmetry groups, potentially limiting scalability.
- The theoretical bounds rely on spectral decay assumptions that may not hold for all kernels or manifolds, and extending to continuous or approximate invariances remains challenging.
Future Work
Future research will focus on extending the framework to continuous and approximate invariances, developing scalable algorithms for large groups, and integrating deep learning-based invariant feature learning. Exploring adaptive methods that learn invariance from data and applying the theory to broader classes of problems such as control and large-scale simulations are promising directions. Additionally, relaxing spectral assumptions and improving computational efficiency will broaden practical applicability.
AI Executive Summary
Bayesian optimization (BO) has become a cornerstone technique for optimizing costly functions across scientific and engineering domains. Despite its success, traditional BO methods often underperform when the target functions exhibit known symmetries or invariances, such as rotational or permutation invariance, which are common in physical systems like nuclear fusion reactors. These symmetries, if properly exploited, can dramatically reduce the number of samples needed to find optimal solutions. However, existing approaches either rely on data augmentation or spatial constraints, which are computationally expensive and lack rigorous theoretical guarantees.
This paper introduces a novel framework that incorporates known group invariances directly into the Gaussian process (GP) kernels used in BO. By constructing invariant kernels under the action of finite groups, the authors develop invariance-aware BO algorithms that leverage the symmetry properties to enhance sample efficiency. The core idea involves defining a totally invariant kernel based on the spectral properties of the group action, which leads to tighter bounds on the maximum information gain—a key quantity dictating sample complexity.
Theoretical analysis demonstrates that the sample complexity decreases linearly with the size of the symmetry group, providing explicit upper bounds on the number of samples required to achieve near-optimality. These bounds are validated through experiments on synthetic functions with permutation and cyclic invariances, showing improvements of up to 3 times faster convergence compared to standard BO. The approach is further applied to the design of a current drive system for a nuclear fusion reactor, where it successfully identified high-performance solutions with significantly fewer evaluations, outperforming traditional methods that failed in this task.
The significance of this work lies in its rigorous theoretical foundation combined with practical effectiveness. It bridges the gap between invariant kernel theory and scalable BO algorithms, offering a systematic way to exploit symmetries in complex optimization problems. Future directions include extending the framework to continuous and approximate invariances, improving computational scalability, and integrating deep learning techniques for invariance learning. Overall, this research marks a substantial step toward more efficient and theoretically grounded optimization in high-dimensional, symmetry-rich applications.
Deep Analysis
Background
贝叶斯优化（BO）在科学与工程中的应用不断深化，尤其在高成本评估场景中表现出色。早期工作如Srinivas等提出的GP-UCB算法奠定了理论基础。近年来，研究者开始关注利用系统的结构信息，如物理知识引导的特征变换和核方法。核不变性理论由Schölkopf等提出，强调通过利用对称性减少样本需求。尽管深度学习中的数据增强能引入不变性，但计算成本高昂。将不变性直接融入核函数，成为提升样本效率的关键路径。此前，有限群作用下的不变核和谱分析已被探索，但在贝叶斯优化中的系统应用尚未充分展开。本文结合群作用的谱性质，提出了理论界限，填补了该领域的空白。
Core Problem
许多实际优化任务具有已知的对称性结构，例如物理系统中的空间对称性或几何变换。传统BO算法未能充分利用这些信息，导致样本需求高、收敛缓慢。现有方法多依赖数据增强或空间约束，存在计算成本高和泛化能力不足的问题。如何在保证理论保证的同时，有效引入不变性，提升样本效率，成为亟待解决的难题。尤其在高维空间和复杂群作用下，优化算法的性能受限，亟需系统的理论分析与实证验证。
Innovation
本研究的创新点主要在于：1）提出基于群作用的完全不变核构造方法，系统引入不变性；2）推导最大信息增益的上界，明确不变性带来的样本效率提升；3）结合Matérn核，兼容高维空间，提供理论保证；4）在合成函数和核聚变优化中验证效果，展示了在高维和大群组情况下的优越性。该方法区别于传统数据增强和空间约束，直接在核层面实现不变性，具有更强的理论支撑和实用价值。
Methodology
- �� 定义目标函数在有限群作用下的完全不变性，构建对应的不变核。• 利用群作用的谱性质，分析核的特征值衰减，推导最大信息增益的界限。• 设计结合Matérn核的贝叶斯优化算法，采用最大方差减少（MVR）和上置信界（UCB）采集策略。• 通过理论分析，证明样本复杂度随群组大小线性下降。• 在合成高维函数和核聚变系统中进行实证验证，比较不变核与非不变核的性能差异。
Experiments
采用合成的群对称函数（如置换群、循环群）进行优化测试，验证不变核在高维（如6D）任务中的样本效率。核聚变系统设计中，利用不变核快速找到高性能方案，减少样本数50%以上。对比基线方法（非不变核、空间约束）展示了显著优势。参数设置包括Matérn核参数、群作用类型和噪声水平，确保结果的普适性。多次重复实验确保统计显著性。
Results
实验证明，不变核在合成任务中提升优化速度2-3倍，尤其在高维和大群组中效果更明显。核聚变应用中，样本需求减少一半以上，找到的方案性能优于传统方法。理论界限验证了样本复杂度随群大小线性下降，支持了算法的有效性。部分不完全不变核在模型偏差存在时仍保持较好性能，显示鲁棒性。
Applications
该方法适用于核聚变、材料设计、机器人控制等需要利用已知对称性的高成本优化场景。只需已知群作用和核函数，即可显著提升样本效率。未来可结合深度学习实现端到端的不变性学习，拓展到更复杂的系统优化。
Limitations & Outlook
依赖于已知的群作用和核的完全不变性，实际中可能存在偏差。大规模群（如对称群）计算成本高，需优化算法。对连续变换和近似不变性支持有限，未来需扩展到更复杂的不变性结构。
Plain Language Accessible to non-experts
想象你在厨房做饭，目标是找到最美味的菜谱。厨房里有很多调料和工具，但无论你把锅放在左边还是右边，味道都一样。这就像目标函数的对称性：只要换个角度，菜的味道没变。传统方法就像每次都试不同位置的调料，浪费很多时间。本文的方法就像提前知道这些调料的对称性，把它们的效果结合在一起，少试几次就能找到最佳菜谱。这样一来，厨师可以用更少的试验，做出更好吃的菜。这个思路也适用于工程设计、核聚变等复杂问题，只要知道一些基本的对称规律，就能大大节省时间和成本。
ELI14 Explained like you're 14
想象你在玩一个游戏，你要找到隐藏的宝藏。游戏地图上有很多隐藏点，但它们其实都遵循一些规则，比如对称或重复。普通的搜索方法就像每次都随机找，花很多时间。现在，假如你知道这些隐藏点的规则，比如它们对称分布，你就可以用更聪明的方法快速找到宝藏。这就像科学家用数学工具，把这些对称的规则加入到搜索策略里，让他们少试几次就能找到最好的方案。这样一来，不仅节省时间，还能解决更复杂的问题，比如设计核聚变反应堆，找到最优的参数组合。这个方法让我们更聪明地利用已知的规律，事半功倍。
Abstract
Bayesian optimisation (BO) is a powerful framework for global optimisation of costly functions, using predictions from Gaussian process models (GPs). In this work, we apply BO to functions that exhibit invariance to a known group of transformations. We show that vanilla and constrained BO algorithms are inefficient when optimising such invariant objectives, and provide a method for incorporating group invariances into the kernel of the GP to produce invariance-aware algorithms that achieve significant improvements in sample efficiency. We derive a bound on the maximum information gain of these invariant kernels, and provide novel upper and lower bounds on the number of observations required for invariance-aware BO algorithms to achieve $ε$-optimality. We demonstrate our method's improved performance on a range of synthetic invariant and quasi-invariant functions. We also apply our method in the case where only some of the invariance is incorporated into the kernel, and find that these kernels achieve similar gains in sample efficiency at significantly reduced computational cost. Finally, we use invariant BO to design a current drive system for a nuclear fusion reactor, finding a high-performance solution where non-invariant methods failed.