LeanAgent: Lifelong Learning for Formal Theorem Proving (2410.06209v7)

Abstract: LLMs have been successful in mathematical reasoning tasks such as formal theorem proving when integrated with interactive proof assistants like Lean. Existing approaches involve training or fine-tuning an LLM on a specific dataset to perform well on particular domains, such as undergraduate-level mathematics. These methods struggle with generalizability to advanced mathematics. A fundamental limitation is that these approaches operate on static domains, failing to capture how mathematicians often work across multiple domains and projects simultaneously or cyclically. We present LeanAgent, a novel lifelong learning framework for formal theorem proving that continuously generalizes to and improves on ever-expanding mathematical knowledge without forgetting previously learned knowledge. LeanAgent introduces several key innovations, including a curriculum learning strategy that optimizes the learning trajectory in terms of mathematical difficulty, a dynamic database for efficient management of evolving mathematical knowledge, and progressive training to balance stability and plasticity. LeanAgent successfully proves 155 theorems previously unproved formally by humans across 23 diverse Lean repositories, many from advanced mathematics. It performs significantly better than the static LLM baseline, proving challenging theorems in domains like abstract algebra and algebraic topology while showcasing a clear progression of learning from basic concepts to advanced topics. In addition, we analyze LeanAgent's superior performance on key lifelong learning metrics. LeanAgent achieves exceptional scores in stability and backward transfer, where learning new tasks improves performance on previously learned tasks. This emphasizes LeanAgent's continuous generalizability and improvement, explaining its superior theorem-proving performance.

• The paper introduces LeanAgent, a lifelong learning framework that applies curriculum learning to rank theorems by an exponential complexity metric.
• It employs dynamic database management and progressive training to mitigate catastrophic forgetting in formal proof development.
• Experiments show an 11-fold performance boost by proving 162 unsolved theorems across diverse Lean repositories, achieving a 94% composite score.

Lifelong Learning in Formal Theorem Proving with LeanAgent

The paper introduces LeanAgent, a novel lifelong learning framework for formal theorem proving, designed to improve automation and efficiency in formalizing mathematical proofs. The framework addresses a critical gap in existing application paradigms of LLMs integrated with interactive proof assistants like Lean, specifically targeting the static nature of previous models which struggle with generalizability across advanced mathematical domains.

Key Innovations

LeanAgent introduces several advancements:

1. Curriculum Learning Strategy: This approach optimizes the learning trajectory by ranking theorems based on their complexity, defined as an exponential function of proof steps ($e^S$). This enables the model to build competence progressively from basic to complex mathematical concepts.
2. Dynamic Database Management: The framework manages an evolving repository of mathematical knowledge, allowing efficient data handling and access as LeanAgent navigates through various mathematical proofs.
3. Progressive Training: To address catastrophic forgetting, LeanAgent employs a progressive training regimen for its retriever, allowing continuous adaptation to new proofs while preserving learned knowledge.

Numerical Results

LeanAgent’s performance is significantly enhanced, showcasing its capability to prove 162 previously unsolved theorems across 23 diverse Lean repositories, including domains from abstract algebra to algebraic topology. Notably, it achieves up to an 11-fold improvement over a baseline static LLM model.

Analysis of Lifelong Learning Metrics

LeanAgent excels in lifelong learning metrics such as stability and backward transfer, indicating its robust capability to integrate new knowledge without degrading previously acquired skills. The model's near-perfect composite score of 94% reflects its adeptness in balancing stability and plasticity, maintaining high theorem-proving efficacy throughout learning epochs.

Theoretical and Practical Implications

The practical implications of LeanAgent are profound, offering a tool for mathematicians to formalize challenging theorems across multiple domains dynamically. Theoretically, it demonstrates an effective blend of curriculum learning and progressive training that can be employed in more complex AI tasks. LeanAgent’s approach can potentially be extrapolated to other formal reasoning tasks and integrated with systems like Lean Copilot for real-time formalization assistance.

Future Research Directions

Further research could focus on integrating reinforcement learning techniques for synthetic data generation, enhancing the curriculum’s adaptability. Additionally, exploring LeanAgent's adaptability in domains with scarce data could yield insights into its robustness and versatility.

LeanAgent represents a substantial stride in leveraging LLMs for formal theorem proving, illustrating the potential of lifelong learning frameworks to transform complex reasoning tasks in artificial intelligence.