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Bayesian Optimization on Networks

Wenwen Li, Daniel Sanz-Alonso, Ruiyi Yang

TL;DR

This work addresses global optimization of expensive black-box objectives defined on networks modeled as compact metric graphs. It introduces geometry-aware Bayesian optimization using Whittle–Matérn Gaussian processes defined via SPDEs on graphs, and develops practical finite-element kernel representations to enable scalable updates. Theoretical contributions include regret bounds for IGP-UCB and GP-TS in both an idealized exact-kernel setting and a misspecified FEM setting with unknown smoothness; experiments on synthetic graphs and a telecommunication network inversion problem demonstrate the superiority of graph-consistent kernels over Euclidean kernels. The results substantiate the value of incorporating network geometry into GP surrogates for efficient optimization and robust kernel-misspecification handling in practice.

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-Matérn 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-Matérn 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.

Bayesian Optimization on Networks

TL;DR

This work addresses global optimization of expensive black-box objectives defined on networks modeled as compact metric graphs. It introduces geometry-aware Bayesian optimization using Whittle–Matérn Gaussian processes defined via SPDEs on graphs, and develops practical finite-element kernel representations to enable scalable updates. Theoretical contributions include regret bounds for IGP-UCB and GP-TS in both an idealized exact-kernel setting and a misspecified FEM setting with unknown smoothness; experiments on synthetic graphs and a telecommunication network inversion problem demonstrate the superiority of graph-consistent kernels over Euclidean kernels. The results substantiate the value of incorporating network geometry into GP surrogates for efficient optimization and robust kernel-misspecification handling in practice.

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-Matérn 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-Matérn 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.

Paper Structure

This paper contains 16 sections, 5 theorems, 53 equations, 1 figure, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Outline and Main Contributions
  4. Notation
  5. Bayesian Optimization on Metric Graphs
  6. Problem Statement
  7. Gaussian Processes and Bayesian Optimization
  8. Choice of Kernel: Whittle-Matérn Gaussian Processes
  9. Regret Bounds
  10. Bayesian Optimization with Finite Element Kernel Representation
  11. Finite Element Kernels
  12. FEM space on $\Gamma$
  13. FEM Approximation
  14. Rational Approximation
  15. Regret Bounds
  16. ...and 1 more sections

Key Result

Lemma 2.1

Suppose the $\{\psi_i\}_{i=1}^{\infty}$ are $L^2(\Gamma)$ normalized, then $\operatorname{sup}_i \|\psi_i\|_{L^\infty(\Gamma)} \leq \Psi$ for a constant $\Psi$ independent of $i$. As a consequence, for $\alpha>\frac{1}{4}$, $|k(x,x')|\leq \overline{k}$ for some constant $\overline{k}<\infty$. Moreov where $S= \sum_{i=1}^\infty \lambda_i^{-2\alpha+1/2}<\infty$ and we recall $d$ is the shortest path

Figures (1)

  • Figure 1: Telecommunication network modeled as a compact metric graph (left) and a sample from the Whittle-Matérn GP defined in Subsection \ref{['sec:exact kernel']} (right).

Theorems & Definitions (7)

  • Lemma 2.1
  • Theorem 2.2
  • proof
  • Lemma 3.1
  • Lemma 3.2
  • Theorem 3.3
  • proof : Proof of Theorem \ref{['thm:regret fem']}