Bayesian Optimization with Exponential Convergence
Proposes a Bayesian optimization method with exponential convergence, avoiding auxiliary optimization and δ-cover sampling.
Key Findings
Methodology
This paper introduces a GP-based Bayesian optimization algorithm leveraging an unknown semi-metric `, combined with a multi-candidate boundary strategy to eliminate reliance on non-convex global optimization and δ-cover sampling. The algorithm employs hierarchical partitioning to maintain hyperrectangles, using GP posterior mean and uncertainty to guide sampling. It incorporates a set of infinitely many bounds, dynamically selecting promising regions via a dual filtering mechanism based on GP's confidence bounds and the unknown semi-metric `. Theoretical analysis proves exponential convergence under smoothness and boundary assumptions, surpassing prior methods dependent on impractical sampling. The approach balances exploration and exploitation efficiently, with rigorous regret bounds.
Key Results
- Experimental evaluations on benchmark functions such as Rosenbrock, Hartmann, and Shekel demonstrate that IMGPO achieves faster convergence, with significantly fewer evaluations (e.g., in Sin1, IMGPO takes 1.61 seconds vs. BaMSOO's 43.80 seconds). It maintains exponential regret decay across dimensions, including high-dimensional cases like Sin1000, validating the theoretical guarantees.
- The method outperforms existing algorithms like BaMSOO, GP-PI, and GP-EI in terms of convergence speed and computational efficiency, especially under limited evaluation budgets. The experiments confirm the effectiveness of the infinite boundary strategy in practical scenarios.
- Results show that the proposed approach consistently reduces simple regret exponentially, demonstrating robustness and scalability, with theoretical bounds matching empirical performance.
Significance
This work advances Bayesian optimization by providing a theoretically grounded, practical algorithm with exponential convergence guarantees, addressing longstanding limitations related to sampling strategies and boundary tightness. It broadens the applicability of Bayesian methods to large-scale, high-dimensional problems in machine learning, engineering, and biological modeling, where evaluation costs are high. The elimination of auxiliary optimization steps simplifies implementation and reduces computational overhead, making it suitable for real-world applications requiring rapid convergence within limited budgets.
Technical Contribution
The key technical innovation lies in integrating an infinite set of candidate bounds with GP posterior information, enabling the algorithm to adaptively select promising regions without relying on δ-cover sampling. The hierarchical partitioning combined with dual filtering based on unknown semi-metrics and GP confidence bounds yields a new theoretical framework that guarantees exponential regret decay. The analysis extends existing regret bounds, providing explicit dependence on smoothness parameters and problem dimension, and demonstrating improved convergence rates over prior methods like BaMSOO.
Novelty
This is the first Bayesian optimization algorithm achieving exponential convergence without auxiliary non-convex optimization or δ-cover sampling. The core novelty is the use of an infinite candidate boundary set, leveraging GP posterior information to adaptively refine search regions, which significantly improves theoretical guarantees and practical efficiency compared to previous approaches relying on fixed bounds or hierarchical partitioning alone.
Limitations
- The convergence rate deteriorates with increasing dimensionality, as the factor λ approaches 1, indicating scalability challenges in very high dimensions.
- The method relies on assumptions of smoothness and the existence of an unknown semi-metric, which may not hold for highly irregular or discontinuous functions.
- Hyperparameter sensitivity and computational costs associated with GP hyperparameter updates may limit real-time applications in extremely large-scale problems.
Future Work
Future research will focus on extending the framework to high-dimensional settings via sparse or deep Gaussian processes, integrating adaptive boundary strategies, and exploring multi-objective optimization scenarios. Additionally, developing more robust boundary estimation techniques and reducing computational overhead for hyperparameter tuning will be key directions.
AI Executive Summary
This paper introduces a novel Bayesian optimization framework that guarantees exponential convergence without relying on auxiliary non-convex optimization or δ-cover sampling, longstanding hurdles in the field. Traditional methods like GP-UCB and BaMSOO, while effective, depend heavily on sampling strategies that are computationally demanding and theoretically restrictive. The proposed approach leverages an innovative infinite candidate boundary strategy, combined with hierarchical partitioning and Gaussian process modeling, to efficiently narrow down the search space.
The core idea is to incorporate an unknown semi-metric `, which encodes the function's continuity properties, into a dual filtering process that considers all possible bounds simultaneously. This allows the algorithm to adaptively focus on promising regions, balancing exploration and exploitation dynamically. The theoretical analysis confirms that, under mild smoothness assumptions, the method achieves exponential regret decay, outperforming prior algorithms both in theory and practice.
Extensive experiments on benchmark functions such as Rosenbrock, Hartmann, and high-dimensional Sin1000 demonstrate the method's superior convergence speed and computational efficiency. The results validate the theoretical guarantees, showing that the algorithm converges rapidly within limited evaluations, making it suitable for real-world applications where evaluation costs are high.
This work significantly advances the state-of-the-art in Bayesian optimization, providing a practical, scalable, and theoretically sound tool for complex black-box optimization problems across diverse fields. Future directions include high-dimensional extensions, multi-objective scenarios, and more robust boundary estimation techniques, promising broader impact and deeper understanding of global optimization challenges.
Deep Dive
Abstract
This paper presents a Bayesian optimization method with exponential convergence without the need of auxiliary optimization and without the delta-cover sampling. Most Bayesian optimization methods require auxiliary optimization: an additional non-convex global optimization problem, which can be time-consuming and hard to implement in practice. Also, the existing Bayesian optimization method with exponential convergence requires access to the delta-cover sampling, which was considered to be impractical. Our approach eliminates both requirements and achieves an exponential convergence rate.