Bayesian Optimization on Networks

TL;DR

This paper introduces Bayesian optimization algorithms tailored for network-structured problems modeled as metric graphs, using Gaussian process surrogates that respect the network geometry, with theoretical guarantees and practical demonstrations.

Contribution

It develops network-aware Bayesian optimization methods with regret bounds and finite element approximations, applied to synthetic and real telecommunication network problems.

Findings

Algorithms effectively optimize benchmark functions on metric graphs.

Finite element approach handles unknown smoothness of objectives.

Numerical results show successful Bayesian inversion on a telecommunication network.

Abstract

This paper studies optimization on networks modeled as metric graphs. Motivated by applications where the objective function is expensive to evaluate or only available as a black box, we develop Bayesian optimization algorithms that sequentially update a Gaussian process surrogate model of the objective to guide the acquisition of query points. To ensure that the surrogates are tailored to the network's geometry, we adopt Whittle-Mat\'ern Gaussian process prior models defined via stochastic partial differential equations on metric graphs. In addition to establishing regret bounds for optimizing sufficiently smooth objective functions, we analyze the practical case in which the smoothness of the objective is unknown and the Whittle-Mat\'ern prior is represented using finite elements. Numerical results demonstrate the effectiveness of our algorithms for optimizing benchmark objective functions on a synthetic metric graph and for Bayesian inversion via maximum a posteriori estimation on a telecommunication network.