Bayesian Optimization (BO) is an effective framework for globally optimizing functions whose evaluations are expensive. It is particularly effective for optimizing functions defined over continuous domains and explicitly handles stochastic noise in evaluations. As a result, it is widely applied in areas such as hyperparameter tuning, robotics policy search, and scientific experiment design, where sample efficiency is essential. Its two-step procedure consists of model fitting followed by optimization of the acquisition function, which is often treated as a generic black-box problem despite its structured nature. In this work, we introduce KSOS-BO, a kernel-based derivative-free framework for BO acquisition optimization. KSOS-BO formulates the optimization of the acquisition function as a semidefinite program with kernel-induced representations, enabling a structured global search. Across a diverse set of benchmark functions with varying landscape properties, KSOS-BO consistently outperforms derivative-free baselines using Sobol Search, Differential Evolution, or CMA-ES to optimize the acquisition function, achieving an average regret improvement of 81.16% on 10/15 benchmarks. In particular, KSOS-BO demonstrates strong performance in highly multimodal and unimodal but ill-conditioned functions, indicating its applicability to diverse landscape structures. Despite a higher per-iteration computational cost, it converges faster in wall-clock time with an average improvement of 93.55% on 10/15 benchmarks, as it reaches high-quality solutions with fewer evaluations. Limitations include reduced effectiveness on functions with steep drops or plate-shaped regions.

翻译：贝叶斯优化（BO）是一种高效优化评估代价高昂的全局函数的框架，尤其适用于连续域上的函数优化，并能显式处理评估中的随机噪声。因此，它在超参数调优、机器人策略搜索和科学实验设计等对样本效率要求极高的领域得到广泛应用。其两步骤流程包括模型拟合与采集函数优化——尽管采集函数具有结构化特性，但常被当作通用黑箱问题处理。本文提出KSOS-BO，一种基于核的无导数BO采集优化框架。KSOS-BO将采集函数优化问题转化为具有核诱导表示的半定规划问题，从而实现结构化全局搜索。在具有不同地形特征的多样化基准函数集上，KSOS-BO在优化采集函数时始终优于使用Sobol搜索、差分进化或CMA-ES的无导数基准方法，在10/15个基准上平均遗憾度提升81.16%。特别地，KSOS-BO在高多模态和单峰但病态函数上表现优异，证明了其对多样化地形结构的适用性。尽管单次迭代计算成本较高，但通过更少的评估达到高质量解，其在挂钟时间上收敛更快，在10/15个基准上平均提升93.55%。局限性包括在陡降或平板区域函数上的有效性降低。