This paper integrates manifold learning techniques within a \emph{Gaussian process upper confidence bound} algorithm to optimize an objective function on a manifold. Our approach is motivated by applications where a full representation of the manifold is not available and querying the objective is expensive. We rely on a point cloud of manifold samples to define a graph Gaussian process surrogate model for the objective. Query points are sequentially chosen using the posterior distribution of the surrogate model given all previous queries. We establish regret bounds in terms of the number of queries and the size of the point cloud. Several numerical examples complement the theory and illustrate the performance of our method.

翻译：本文在\emph{高斯过程上置信界}算法框架下融合流形学习技术，以优化流形上的目标函数。我们的方法源于对以下应用场景的考虑：既无法获得完整流形表示，又面临高昂的目标查询代价。我们利用流形样本点云构建目标函数的图高斯过程替代模型，并基于该模型在历史查询结果下的后验分布，依次选取后续查询点。在查询次数和点云规模维度上，我们建立了相应的遗憾界理论。多个数值实验验证了理论分析，并展示了所提方法的实际性能。