Optimization on Manifolds via Graph Gaussian Processes

TL;DR

This paper presents a novel method combining manifold learning and Gaussian process optimization to efficiently optimize functions on manifolds using limited samples and graph-based surrogates.

Contribution

It introduces a graph Gaussian process surrogate model for manifold optimization with theoretical regret bounds and practical numerical demonstrations.

Findings

Regret bounds established for the proposed method

Effective optimization with limited manifold samples

Numerical examples show competitive performance

Abstract

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.