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Global Optimization on Graph-Structured Data via Gaussian Processes with Spectral Representations

Shu Hong, Yongsheng Mei, Mahdi Imani, Tian Lan

TL;DR

The paper addresses optimizing an expensive black-box function defined on the nodes of large graphs with partially observed topology. It introduces a global Bayesian optimization framework that builds a GP surrogate in a learned spectral embedding space by first recovering a low-rank surrogate adjacency $\tilde{A}$ via nuclear-norm regularization, then computing spectral embeddings from $\tilde{A}$, and finally applying GP-based BO with a graph-aware spectral kernel. The authors provide non-asymptotic recovery guarantees under both random and deterministic edge sampling, and develop scalable neural-network parameterizations for online learning of the surrogate and embeddings. Empirical results on synthetic (SBM, RDPG) and real-world (Ego-Facebook) graphs demonstrate faster convergence and improved optimization performance over existing graph BO baselines, highlighting the framework's practical impact for tasks like influence maximization, molecular design, and infrastructure optimization on graphs.

Abstract

Bayesian optimization (BO) is a powerful framework for optimizing expensive black-box objectives, yet extending it to graph-structured domains remains challenging due to the discrete and combinatorial nature of graphs. Existing approaches often rely on either full graph topology-impractical for large or partially observed graphs-or incremental exploration, which can lead to slow convergence. We introduce a scalable framework for global optimization over graphs that employs low-rank spectral representations to build Gaussian process (GP) surrogates from sparse structural observations. The method jointly infers graph structure and node representations through learnable embeddings, enabling efficient global search and principled uncertainty estimation even with limited data. We also provide theoretical analysis establishing conditions for accurate recovery of underlying graph structure under different sampling regimes. Experiments on synthetic and real-world datasets demonstrate that our approach achieves faster convergence and improved optimization performance compared to prior methods.

Global Optimization on Graph-Structured Data via Gaussian Processes with Spectral Representations

TL;DR

The paper addresses optimizing an expensive black-box function defined on the nodes of large graphs with partially observed topology. It introduces a global Bayesian optimization framework that builds a GP surrogate in a learned spectral embedding space by first recovering a low-rank surrogate adjacency via nuclear-norm regularization, then computing spectral embeddings from , and finally applying GP-based BO with a graph-aware spectral kernel. The authors provide non-asymptotic recovery guarantees under both random and deterministic edge sampling, and develop scalable neural-network parameterizations for online learning of the surrogate and embeddings. Empirical results on synthetic (SBM, RDPG) and real-world (Ego-Facebook) graphs demonstrate faster convergence and improved optimization performance over existing graph BO baselines, highlighting the framework's practical impact for tasks like influence maximization, molecular design, and infrastructure optimization on graphs.

Abstract

Bayesian optimization (BO) is a powerful framework for optimizing expensive black-box objectives, yet extending it to graph-structured domains remains challenging due to the discrete and combinatorial nature of graphs. Existing approaches often rely on either full graph topology-impractical for large or partially observed graphs-or incremental exploration, which can lead to slow convergence. We introduce a scalable framework for global optimization over graphs that employs low-rank spectral representations to build Gaussian process (GP) surrogates from sparse structural observations. The method jointly infers graph structure and node representations through learnable embeddings, enabling efficient global search and principled uncertainty estimation even with limited data. We also provide theoretical analysis establishing conditions for accurate recovery of underlying graph structure under different sampling regimes. Experiments on synthetic and real-world datasets demonstrate that our approach achieves faster convergence and improved optimization performance compared to prior methods.

Paper Structure

This paper contains 22 sections, 3 theorems, 13 equations, 3 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Bayesian Optimization on Graphs
  4. Matrix Completion for Graph Structure Recovery
  5. Spectral Embedding for Graph-based Learning
  6. Methodology
  7. Matrix Completion and Spectral Embedding
  8. Rank–constrained Matrix Completion.
  9. Theoretical Guarantees and Sampling Complexity.
  10. Spectral parameterization.
  11. Spectral Graph Drawing for Downstream GP Modeling.
  12. Joint Optimization via Neural Network Representations.
  13. Global Graph Bayesian Optimization using Spectral Embedding
  14. Spectral kernel for Gaussian processes.
  15. GP likelihood and hyperparameter estimation.
  16. ...and 7 more sections

Key Result

Theorem 1

Let $A\in\mathbb{R}^{n\times n}$ be symmetric, $\operatorname{rank}(A)=r$, and $\mu$–incoherent: $\max_i\|Q_{i,:}\|_2^{2}\le\mu r/n$ for the eigen-basis $A=Q \Gamma Q^{\top}$. Sample each entry independently with probability $p=|\Omega|/n^{2}$. There exist constants $C,c,c'>0$ such that if $|\Omega|

Figures (3)

  • Figure 1: Illustration of the proposed global Bayesian optimization (BO) framework on graphs.
  • Figure 2: Comparisons of optimization convergence with 95% confidence intervals.
  • Figure 3: Ablation study on edge sampling method: optimization convergence with 95% confidence intervals.

Theorems & Definitions (5)

  • Theorem 1: Random Sampling
  • Theorem 2: Deterministic Sampling
  • Remark 1: Link to Algorithm \ref{['Alg:edge_sampling']}
  • Lemma 1
  • Remark 2: Different choices of $d_1$ and $d_2$