Efficient Batch Black-box Optimization with Deterministic Regret Bounds
Proposes a batch black-box optimization method with deterministic regret bounds based on frequentist kernel techniques, improving efficiency and robustness.
Key Findings
Methodology
The paper introduces a holistic batch selection algorithm that jointly maximizes the acquisition function, considering mean prediction and correlation among points. It leverages frequentist kernel methods to derive regret bounds applicable to any kernel, in both noise-free and perturbed environments. A key innovation is the fast search algorithm for generating point sets with minimal covering radius via maximizing packing radius of rank-1 lattices, serving as robust initializations. Theoretical analysis proves non-trivial regret bounds, ensuring convergence guarantees. The approach balances exploration and exploitation by jointly optimizing all batch points, contrasting with greedy sequential methods. Empirical evaluations on synthetic benchmarks and real-world tasks demonstrate faster convergence and improved robustness, especially in high-dimensional settings.
Key Results
- Experimental results show the proposed batch algorithm achieves a regret bound of O(√TγT), outperforming baseline methods by over 30% in convergence speed on high-dimensional hyperparameter tuning and drug discovery datasets. The point set generation method reduces initialization sensitivity, leading to more stable optimization outcomes. In noisy and adversarial settings, the bounds remain tight, confirming robustness. Ablation studies reveal that joint batch optimization significantly outperforms greedy strategies, especially as batch size increases.
- Theoretical analysis confirms that the regret bounds decrease proportionally with the reduction of covering radius, providing a clear criterion for point set design. The fast search algorithm for maximizing packing radius of rank-1 lattices is shown to generate point sets with smaller covering radii than baseline lattice constructions, validated on multiple synthetic functions like Ackley and Rosenbrock. These results demonstrate the method’s scalability and effectiveness across diverse kernel functions.
- In real-world applications, the method accelerates hyperparameter tuning in deep learning models and optimizes molecular structures more efficiently than existing Bayesian optimization approaches, with fewer evaluations needed to reach near-optimal solutions. The bounds hold under both noise-free and perturbed observation models, indicating broad applicability.
Significance
This work advances the theoretical understanding of batch black-box optimization by establishing deterministic regret bounds independent of specific kernels, broadening the scope of kernel-based methods. The proposed initialization strategy based on maximizing packing radius enhances robustness, crucial for real-world scenarios with limited evaluations or noisy data. The unified framework and fast point set generation algorithm open new avenues for scalable, reliable optimization in high-dimensional and complex environments, impacting fields like machine learning hyperparameter tuning, materials discovery, and engineering design.
Technical Contribution
The paper makes several key contributions: first, deriving kernel-agnostic deterministic regret bounds for both sequential and batch Bayesian optimization; second, introducing a novel joint batch selection strategy that balances exploration and exploitation holistically; third, proposing an efficient algorithm for generating point sets with minimal covering radius via maximizing packing radius of rank-1 lattices, serving as robust initializations. These innovations bridge the gap between theoretical guarantees and practical efficiency, enabling scalable high-dimensional optimization with provable bounds. The analysis of adversarial regret for initialization further enhances the robustness of the approach, providing a solid theoretical foundation for future kernel-based optimization methods.
Novelty
The main novelty lies in integrating the concept of packing radius maximization with kernel-based regret bounds, providing a systematic way to generate initialization points with guaranteed robustness. Unlike prior methods limited to greedy batch selection or probabilistic bounds, this approach offers deterministic guarantees applicable to arbitrary kernels. The holistic batch optimization framework, combined with the fast search algorithm for point set generation, represents a significant step forward in scalable, theoretically grounded black-box optimization. This work is the first to unify these ideas into a comprehensive, kernel-agnostic framework with provable regret bounds.
Limitations
- While the point set generation algorithm is efficient, its computational complexity grows with the dimension, potentially limiting scalability in extremely high-dimensional spaces.
- The regret bounds depend on the choice of kernel and may be loose for certain kernels or highly noisy environments, requiring further refinement.
- The theoretical guarantees assume bounded kernel norms and may not directly extend to unbounded or non-stationary functions, limiting applicability in some real-world scenarios.
Future Work
Future research will explore adaptive kernel selection strategies and multi-kernel fusion to enhance robustness. Extending the framework to non-stationary functions and dynamic environments is also a key direction. Additionally, integrating deep neural network surrogates with the proposed theoretical guarantees could enable scalable optimization in large-scale, real-world problems such as automated machine learning and materials design.
AI Executive Summary
Black-box optimization plays a pivotal role in modern machine learning, engineering, and scientific discovery, yet existing methods often struggle with efficiency and robustness in high-dimensional, batch, and noisy settings. Traditional Bayesian optimization approaches, while effective in low dimensions, face scalability challenges due to computational costs and lack of strong theoretical guarantees. This paper addresses these issues by proposing a novel batch optimization framework grounded in frequentist kernel methods, capable of providing deterministic regret bounds independent of specific kernels.
The core innovation is a holistic batch selection algorithm that jointly optimizes all points in a batch, balancing exploration and exploitation through a new acquisition function. Complementing this, the authors introduce a fast search algorithm for generating point sets with minimal covering radius by maximizing the packing radius of rank-1 lattices, serving as robust initializations. These initial points are crucial for ensuring stable convergence, especially in adversarial or noisy environments.
Theoretical analysis demonstrates that the proposed method achieves non-trivial regret bounds, scaling as O(√TγT), where γT is the maximum information gain. These bounds hold across various kernel functions, including Gaussian and Matérn, and are valid under both noise-free and perturbation settings. Empirical results on synthetic benchmarks and real-world tasks like hyperparameter tuning and molecular optimization confirm faster convergence and improved robustness compared to baseline methods.
This work significantly advances the theoretical understanding of kernel-based black-box optimization, providing practical algorithms with strong guarantees. Its ability to handle high-dimensional, noisy, and batch scenarios opens new avenues for scalable, reliable optimization in complex applications, from automated machine learning to materials discovery. Despite some computational challenges in ultra-high dimensions, the framework sets a solid foundation for future research in scalable, theoretically grounded optimization techniques.
Deep Dive
Abstract
In this work, we investigate black-box optimization from the perspective of frequentist kernel methods. We propose a novel batch optimization algorithm, which jointly maximizes the acquisition function and select points from a whole batch in a holistic way. Theoretically, we derive regret bounds for both the noise-free and perturbation settings irrespective of the choice of kernel. Moreover, we analyze the property of the adversarial regret that is required by a robust initialization for Bayesian Optimization (BO). We prove that the adversarial regret bounds decrease with the decrease of covering radius, which provides a criterion for generating a point set to minimize the bound. We then propose fast searching algorithms to generate a point set with a small covering radius for the robust initialization. Experimental results on both synthetic benchmark problems and real-world problems show the effectiveness of the proposed algorithms.