Bayesian optimization is highly effective for optimizing expensive-to-evaluate black-box functions, but it faces significant computational challenges due to the cubic per-iteration cost 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 a novel algorithm, Bayesian optimization by kernel regression and density-based exploration (BOKE). 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 under noisy evaluations. 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 and several other baseline 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：贝叶斯优化对于优化评估代价高昂的黑盒函数非常有效，但由于高斯过程每次迭代的立方复杂度，导致其总时间复杂度随迭代次数呈四次方增长，面临显著的计算挑战。为克服这一局限，我们提出一种新型算法——基于核回归与密度探索的贝叶斯优化（BOKE）。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，并将二者整合到置信界准则中以引导优化过程，从而将计算复杂度降至二次方。我们的理论分析严格证明了BOKE在含噪评估条件下的全局收敛性。通过在合成任务和实际优化任务上的大量数值实验，我们证明BOKE不仅与基于高斯过程的方法及其他若干基线方法相比具有竞争力，而且展现出卓越的计算效率。这些结果凸显了BOKE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方案。