Bayesian optimization is highly effective for optimizing expensive-to-evaluate black-box functions, but it faces significant computational challenges due to the high computational complexity of Gaussian processes, which results in a total time complexity that is quartic with respect to the number of iterations. To address this limitation, we propose the Bayesian Optimization by Kernel regression and density-based Exploration (BOKE) algorithm. BOKE uses kernel regression for efficient function approximation, kernel density for exploration, and integrates them into the confidence bound criteria to guide the optimization process, thus reducing computational costs to quadratic. Our theoretical analysis rigorously establishes the global convergence of BOKE and ensures its robustness in noisy settings. Through extensive numerical experiments on both synthetic and real-world optimization tasks, we demonstrate that BOKE not only performs competitively compared to Gaussian process-based methods but also exhibits superior computational efficiency. These results highlight BOKE's effectiveness in resource-constrained environments, providing a practical approach for optimization problems in engineering applications.

翻译：贝叶斯优化在评估代价高昂的黑箱函数优化中极为有效，但其面临显著的计算挑战，主要源于高斯过程的高计算复杂度，导致总时间复杂度随迭代次数呈四次方增长。为应对这一局限，本文提出基于核回归与密度探索的贝叶斯优化算法。该算法采用核回归实现高效函数逼近，利用核密度进行探索，并将二者集成至置信区间准则中以引导优化过程，从而将计算成本降至二次方。理论分析严格证明了该算法的全局收敛性，并确保其在噪声环境下的鲁棒性。通过在合成与真实世界优化任务上的大量数值实验，我们证明该算法不仅性能与基于高斯过程的方法相当，还展现出更优的计算效率。这些结果凸显了该算法在资源受限环境中的有效性，为工程应用中的优化问题提供了实用解决方案。