PatternBoost: Constructions in Mathematics with a Little Help from AI (2411.00566v1)

Abstract: We introduce PatternBoost, a flexible method for finding interesting constructions in mathematics. Our algorithm alternates between two phases. In the first local'' phase, a classical search algorithm is used to produce many desirable constructions. In the secondglobal'' phase, a transformer neural network is trained on the best such constructions. Samples from the trained transformer are then used as seeds for the first phase, and the process is repeated. We give a detailed introduction to this technique, and discuss the results of its application to several problems in extremal combinatorics. The performance of PatternBoost varies across different problems, but there are many situations where its performance is quite impressive. Using our technique, we find the best known solutions to several long-standing problems, including the construction of a counterexample to a conjecture that had remained open for 30 years.

• The paper demonstrates a novel iterative method that hybridizes greedy local search with transformer-based global pattern recognition to solve complex mathematical constructions.
• The methodology leverages a feedback loop between local optimization and global pattern learning to produce near-optimal solutions in extremal combinatorics.
• The research paves the way for versatile AI-assisted approaches that simplify addressing longstanding mathematical challenges across various domains.

Overview of "PatternBoost: Constructions in Mathematics with a Little Help from AI"

This paper presents "PatternBoost," an innovative approach that leverages machine learning, specifically transformer architectures, in conjunction with classical search algorithms to solve complex mathematical construction problems. The authors propose an iterative method that combines local optimization with global pattern recognition, utilizing transformers to generate and refine mathematical constructions. This method is particularly applied to extremal combinatorics, providing a new toolset for mathematicians seeking solutions to longstanding problems.

The PatternBoost Methodology

The main concept behind PatternBoost is a two-phase iterative process:

1. Local Phase: A classical search algorithm, typically a greedy method, is employed to generate a set of initial constructions. These constructions are evaluated and filtered based on a bespoke scoring function appropriate to the mathematical problem at hand.
2. Global Phase: A transformer neural network is trained on the top constructions from the local search. The trained model generates new candidate constructions by recognizing global patterns from the input dataset. The results of these suggestions are used as seeds for the subsequent local phase, creating a feedback loop that iteratively enhances the quality of solutions.

The strength of PatternBoost lies in its ability to exploit the rich, abstract representations offered by transformers, aiming to capture subtle patterns and structures that are challenging to identify through human intuition or isolated local search strategies.

Applications and Results

The paper details several applications of PatternBoost in extremal combinatorics. It successfully finds optimal or near-optimal solutions to longstanding open problems, including the construction of counterexamples to conjectures that remained unresolved for decades. Notably, the paper highlights its role in generating mathematical structures that produce competitive results compared to traditional handcrafted solutions. While the method does not always surpass human experts in all domains, it provides a robust, general-purpose tool that significantly narrows the gap.

Technical Implications

PatternBoost's hybrid approach can be seen as an evolution of genetic algorithms, where the neural network acts to enrich the generation pool based on an understanding of the problem's global constraints. This method's adaptability to various problems without the need for intricate machine learning expertise democratizes its utilization among mathematicians.

Notably, the paper underscores the model's resilience to operate "off the shelf," simplifying its deployment in mathematical research without necessitating extensive parameter tuning or computational resources beyond what's typically available in academic environments.

Future Directions

The research opens several avenues for future exploration:

• Scalability to Other Domains: While demonstrated on combinatorial problems, PatternBoost could potentially apply to other mathematical or even non-mathematical domains where construction or optimization problems arise.
• Enhanced Local Search Strategies: Further refinement of local search algorithms in tandem with transformer outputs could enhance results.
• Tokenization and Data Representation: Exploring more sophisticated tokenization methods could improve model learning efficiencies, particularly for complex or higher-dimensional problems.
• Interdisciplinary Applications: The applicability of PatternBoost in scientific domains where both pattern recognition and optimization are critical could lead to significant interdisciplinary breakthroughs.

In conclusion, this work represents a significant advancement in applying AI techniques to mathematical construction problems, offering new possibilities and insights into problem-solving methodologies that bridge human intuition and machine learning capabilities.