Bayesian optimization (BO) is a sample-efficient method and has been widely used for optimizing expensive black-box functions. Recently, there has been a considerable interest in BO literature in optimizing functions that are affected by context variable in the environment, which is uncontrollable by decision makers. In this paper, we focus on the optimization of functions' expectations over continuous context variable, subject to an unknown distribution. To address this problem, we propose two algorithms that employ kernel density estimation to learn the probability density function (PDF) of continuous context variable online. The first algorithm is simpler, which directly optimizes the expectation under the estimated PDF. Considering that the estimated PDF may have high estimation error when the true distribution is complicated, we further propose the second algorithm that optimizes the distributionally robust objective. Theoretical results demonstrate that both algorithms have sub-linear Bayesian cumulative regret on the expectation objective. Furthermore, we conduct numerical experiments to empirically demonstrate the effectiveness of our algorithms.

翻译：贝叶斯优化作为一种样本高效的方法，已被广泛用于优化昂贵的黑箱函数。近年来，贝叶斯优化领域对优化受环境中不可控上下文变量影响的函数产生了浓厚兴趣。本文聚焦于在连续上下文变量服从未知分布的条件下，优化函数对该变量的期望值。为解决此问题，我们提出两种算法，采用核密度估计在线学习连续上下文变量的概率密度函数。第一种算法较为简单，直接基于估计的概率密度函数优化期望值。考虑到当真实分布复杂时，估计的概率密度函数可能存在较大误差，我们进一步提出第二种算法，用于优化分布鲁棒目标。理论结果表明，两种算法在期望目标上的贝叶斯累积遗憾均呈亚线性增长。此外，我们通过数值实验验证了算法的有效性。