Optimization on Manifolds via Graph Gaussian Processes
This work addresses optimization challenges in domains like robotics or physics where data is limited and manifolds are complex, representing an incremental improvement by combining existing techniques.
The paper tackles the problem of optimizing expensive-to-query objective functions on manifolds without a full representation, by integrating manifold learning into a Gaussian process upper confidence bound algorithm using a graph Gaussian process surrogate model, and establishes regret bounds with numerical examples to illustrate performance.
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.