BoGrape: Bayesian optimization over graphs with shortest-path encoded

TL;DR

BoGrape tackles optimization over graphs for expensive black-box objectives by marrying Bayesian optimization with shortest-path graph kernels encoded into a mixed-integer program. It introduces four variants of shortest-path kernels (SP, SSP, ESP, ESSP) and develops a complete MIP encoding for shortest paths and kernel evaluations, enabling global optimization of the BO acquisition function under mixed graph-feature domains. The approach uses a Gaussian-process surrogate with a graph-kernel prior and optimizes the acquisition via PK-MIQP, providing theoretical groundwork for global optimality in graph search. Experimental results on QM7 and QM9 molecular design tasks demonstrate competitive predictive performance and clear BO advantages over feasible baselines, with SSP often excelling in larger graphs due to model simplicity and efficiency. The work provides a flexible, constraint-aware framework for graph-domain BO with potential applications in neural architecture search and sensor placement.

Abstract

Graph-structured data play an important role across science and industry. This paper asks: how can we optimize over graphs, for instance to find the best graph structure and/or node features that minimize an expensive-to-evaluate black-box objective? Such problem settings arise, e.g., in molecular design, neural architecture search, and sensor placement. Bayesian optimization is a powerful tool for optimizing black-box functions, and existing technologies can be applied to optimize functions over nodes of a single fixed graph. We present Bayesian optimization acquisition functions for a class of shortest-path kernels and formulate them as mixed-integer optimization problems, enabling global exploration of the graph domain while maintaining solution feasibility when problem-specific constraints are present. We demonstrate our proposed approach, BoGrape, on several molecular design case studies.

Figures (5)

• Figure 1: Key components of BoGrape. The graph kernel comprises $k_G$ and $k_F$ on the graph and feature levels, respectively. The graph GP is subsequently trained using the chosen kernel and samples. GP posterior information is used to build the acquisition function, e.g., LCB. Note that graph GP includes discrete graph domains; the continuous domain is only for illustration purposes. The acquisition optimization is formulated as a MIP using the encoding of shortest paths and graph kernels. Solving the MIP gives the next query point.
• Figure 2: Compare predictive performance of GP with different kernels. 100 samples are randomly chosen from the QM7 dataset with various graph sizes, 30 of which are used for training. The predictive mean with one standard deviation (predicted $y$) of the remaining 70 graphs are plotted against their real values (true $y$).
• Figure 3: Bayesian optimization results on QM7 and QM9 with $N\in \{10,20,30\}$. Best objective value is plotted at each iteration. Mean with 0.5 standard deviation over 10 replications is reported.
• Figure 4: Performance of random sampling and Limeade over QM7 and QM9 datasets with different graph size $N$. Simple regret is plotted at each iteration. Mean with 0.5 standard deviation over 10 replications is reported.
• Figure 5: Bayesian optimization results on QM7 and QM9 with $N\in\{15,25\}$. Best objective value is plotted at each iteration. Mean with $0.5$ standard deviation over 10 replications is reported.

Theorems & Definitions (11)

• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Theorem 3.4
• Theorem 3.5
• Remark 3.6
• proof : Proof of Theorem \ref{['thm:SP_fix_size']}