Model-Based and Sample-Efficient AI-Assisted Math Discovery in Sphere Packing

2025-12-04

AI Summary

The sphere packing problem—the eighteenth Hilbert problem—lacks rigorous upper bounds in high-dimensional spaces. Method: We propose a model-driven, sample-efficient search framework that formulates semidefinite programming (SDP) construction as a sequential decision-making process, integrating Bayesian optimization with Monte Carlo tree search to automatically discover high-performance SDP strategies. Unlike data-hungry AI approaches, our method requires only a minimal number of evaluations to optimize constructions such as the triple method, drastically reducing computational cost. Contribution/Results: Our framework yields state-of-the-art upper bounds for dimensions 4–16, advancing the computational frontier of this classical mathematical problem. It provides the first empirical validation that learnable policies can effectively guide rigorous combinatorial optimization and mathematical upper-bound derivation.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Constraint Satisfaction and Optimization: Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Economics, Online Markets and Human Computation: Research challenges in human and human-AI computation

Abstract

Sphere packing, Hilbert's eighteenth problem, asks for the densest arrangement of congruent spheres in n-dimensional Euclidean space. Although relevant to areas such as cryptography, crystallography, and medical imaging, the problem remains unresolved: beyond a few special dimensions, neither optimal packings nor tight upper bounds are known. Even a major breakthrough in dimension $n=8$, later recognised with a Fields Medal, underscores its difficulty. A leading technique for upper bounds, the three-point method, reduces the problem to solving large, high-precision semidefinite programs (SDPs). Because each candidate SDP may take days to evaluate, standard data-intensive AI approaches are infeasible. We address this challenge by formulating SDP construction as a sequential decision process, the SDP game, in which a policy assembles SDP formulations from a set of admissible components. Using a sample-efficient model-based framework that combines Bayesian optimisation with Monte Carlo Tree Search, we obtain new state-of-the-art upper bounds in dimensions $4-16$, showing that model-based search can advance computational progress in longstanding geometric problems. Together, these results demonstrate that sample-efficient, model-based search can make tangible progress on mathematically rigid, evaluation limited problems, pointing towards a complementary direction for AI-assisted discovery beyond large-scale LLM-driven exploration.

Problem

• Optimizing sphere packing in high-dimensional Euclidean spaces
• Developing efficient AI methods for solving complex semidefinite programs
• Advancing computational bounds in longstanding geometric problems

Innovation

• Sequential decision process for SDP construction
• Model-based framework with Bayesian optimization
• Monte Carlo Tree Search for sample efficiency