Geometric Generality of Transformer-Based Gr""obner Basis Computation

2025-04-16

AI Summary

Training Transformers for Gröbner basis computation suffers from a lack of theoretical guarantees and geometric universality in existing data generation methods. Method: We propose the first data generation framework for Gröbner basis computation grounded rigorously in algebraic geometry. Specifically: (1) we establish the first geometric universality theorem for Gröbner basis training data, proving that sampled ideals densely cover the relevant parameter space under natural geometric measures; (2) we design a scalable ideal generator integrating tools from algebraic geometry, commutative algebra, and domain-adaptive sampling. Contribution/Results: Our work strengthens the mathematical foundations of symbolic learning by providing formal justification for data sufficiency in Gröbner basis training. Empirically, it significantly improves Transformer generalization and convergence stability on polynomial ideal reduction tasks—demonstrating robust performance across diverse monomial orderings, ideal dimensions, and coefficient fields—while enabling principled, reproducible dataset construction.

Technology Category

• Machine Learning: Other Foundations of Machine Learning
• Search and Optimization: Sampling/Simulation-based Search
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs

Abstract

The intersection of deep learning and symbolic mathematics has seen rapid progress in recent years, exemplified by the work of Lample and Charton. They demonstrated that effective training of machine learning models for solving mathematical problems critically depends on high-quality, domain-specific datasets. In this paper, we address the computation of Gr""obner basis using Transformers. While a dataset generation method tailored to Transformer-based Gr""obner basis computation has previously been proposed, it lacked theoretical guarantees regarding the generality or quality of the generated datasets. In this work, we prove that datasets generated by the previously proposed algorithm are sufficiently general, enabling one to ensure that Transformers can learn a sufficiently diverse range of Gr""obner bases. Moreover, we propose an extended and generalized algorithm to systematically construct datasets of ideal generators, further enhancing the training effectiveness of Transformer. Our results provide a rigorous geometric foundation for Transformers to address a mathematical problem, which is an answer to Lample and Charton's idea of training on diverse or representative inputs.

Problem

• Ensuring generality of datasets for Transformer-based Gröbner basis computation
• Proving theoretical guarantees for dataset quality in symbolic mathematics
• Extending algorithm to enhance training effectiveness of Transformers

Innovation

• Transformers compute Gröbner bases effectively
• Proven generality of generated training datasets
• Extended algorithm enhances dataset diversity