GAR: Generative Adversarial Reinforcement Learning for Formal Theorem Proving
GAR improves theorem proving efficiency via adversarial RL, achieving a 4.20% gain on MiniF2F-Test.
Key Findings
Methodology
GAR framework combines GANs with RL, optimizing both problem generator and solver through adversarial training. The problem generator creates more challenging problems, while the solver enhances its solving ability. This framework introduces an implicit curriculum learning mechanism, dynamically matching task difficulty with solver capability.
Key Results
- On the MiniF2F-Test benchmark, Goedel-Prover-V2-8B and DeepSeek-Prover-V2-7B achieve an average relative improvement in pass@32 of 4.20%.
- On ProofNet-Test, DeepSeek-Prover-V2's pass@32 increases from 22.58% to 25.81%.
- Adversarial training enables models to tackle more complex problems, enhancing reasoning capabilities.
Significance
The GAR framework holds significant implications for mathematics and computer science. It not only improves theorem proving efficiency but also establishes a general RL paradigm for the co-evolution of problem generation and solving. This approach addresses the inefficiencies of fixed problem sets in traditional methods.
Technical Contribution
GAR overcomes the limitations of existing methods by simultaneously optimizing the problem generator and solver through adversarial training. It introduces an implicit curriculum learning mechanism, dynamically matching problem difficulty with solver capability, thereby enhancing training efficiency and model performance.
Novelty
GAR is the first to introduce generative adversarial reinforcement learning in theorem proving, achieving co-evolution of problem generation and solving by dynamically adjusting problem difficulty. This innovation significantly differs from traditional fixed problem set training methods.
Limitations
- GAR may struggle with extremely complex theorems as the problem generator might fail to generate appropriately challenging problems.
- Due to the complexity of adversarial training, it requires significant computational resources and time.
Future Work
Future research directions include optimizing the problem generator's generation capabilities, reducing training time, and applying GAR to other reasoning-intensive domains.
AI Executive Summary
The ability to solve mathematical problems has long been considered a hallmark of human intelligence and a key objective of machine learning. Existing methods often rely on expensive online reinforcement learning or expert iteration, but these approaches depend on fixed problem sets, leading to inefficient training and difficulty in tackling complex problems.
To address these issues, this paper proposes GAR: a generative adversarial reinforcement learning framework. This framework optimizes both the problem generator and solver through adversarial training, dynamically adjusting problem difficulty to match the solver's capabilities. Experiments show that models trained with GAR perform exceptionally well on the MiniF2F-Test and ProofNet-Test benchmarks, demonstrating its effectiveness and generality.
GAR not only makes significant progress in formal proving but also establishes a general RL paradigm for the co-evolution of problem generation and solving in verifiable environments. Future research will focus on optimizing training efficiency and exploring its application in other reasoning-intensive domains.
Deep Analysis
Background
Formal mathematical reasoning has been a key focus of AI research. With the advent of large language models, developing accurate and reliable reasoning systems has become a research hotspot. However, existing methods face challenges in handling complex mathematical problems, especially in scenarios requiring formal verification.
Core Problem
Existing theorem proving methods rely on fixed problem sets, leading to inefficient training and difficulty in solving complex problems. A key challenge is dynamically adjusting problem difficulty to match the solver's capabilities.
Innovation
The GAR framework optimizes both the problem generator and solver through adversarial training, dynamically adjusting problem difficulty. Its innovation lies in introducing an implicit curriculum learning mechanism, matching problem difficulty with solver capability.
Methodology
- �� Problem generator creates more challenging problems.
- �� Solver attempts to solve generated problems.
- �� Adversarial training: problem generator and solver optimize each other.
- �� Implicit curriculum learning mechanism introduced.
Experiments
Experiments were conducted on the MiniF2F-Test and ProofNet-Test benchmarks, using DeepSeek-Prover-V2 and Goedel-Prover-V2 as baseline models. Through adversarial training, the models performed exceptionally well on multiple benchmarks, validating the effectiveness of GAR.
Results
On MiniF2F-Test, models trained with GAR performed excellently, with Goedel-Prover-V2-8B's pass@32 increasing to 80.33%. On ProofNet-Test, DeepSeek-Prover-V2's pass@32 increased from 22.58% to 25.81%.
Applications
The GAR framework can be applied to formal mathematical proving, automated reasoning systems, and other domains requiring dynamic problem difficulty adjustment.
Limitations & Outlook
GAR may struggle with extremely complex theorems, and the training process is resource-intensive. Future research will focus on optimizing training efficiency and exploring applications in other domains.
Plain Language Accessible to non-experts
Imagine a school where the teacher adjusts the course difficulty based on the students' learning progress. GAR is like this teacher, dynamically adjusting problem difficulty based on the solver's capability, allowing the solver to gradually improve and eventually solve more complex problems.
ELI14 Explained like you're 14
Imagine you're playing a game where every time you get stronger, the game gets harder. That's what GAR does! It adjusts the problem difficulty based on the solver's ability, helping the solver improve and eventually tackle more complex math problems.
Glossary
Generative Adversarial Network (GAN)
A model that generates data through adversarial training.
Used to generate more challenging problems.
Reinforcement Learning (RL)
A method of training models through a reward mechanism.
Used to train the solver to improve its problem-solving ability.
Implicit Curriculum Learning
A mechanism that dynamically adjusts learning task difficulty.
Used to match the solver's capability.
MiniF2F-Test
A benchmark for evaluating theorem proving capabilities.
Used to validate the effectiveness of GAR.
ProofNet-Test
A benchmark for advanced mathematical theorem proving.
Used to assess GAR's performance on complex problems.
Open Questions Unanswered questions from this research
- 1 How to further optimize the problem generator's generation capabilities to reduce training time and resource consumption.
- 2 The potential and challenges of applying GAR in other reasoning-intensive domains.
Applications
Immediate Applications
Automated Theorem Proving
Improves theorem proving efficiency and accuracy by dynamically adjusting problem difficulty.
Long-term Vision
Intelligent Education Systems
Dynamically adjusts course content based on students' learning progress to enhance learning outcomes.
Abstract
Solving math problems through verifiable languages such as Lean has significantly impacted both the mathematics and computer science communities. Current state-of-the-art models are often trained with expensive online Reinforcement Learning (RL) or expert iteration. However, these approaches rely on fixed problem sets, which causes inefficient training and limits the model to tackle complex problems. To overcome these limitations, we propose **GAR**: *Generative Adversarial Reinforcement learning*, a comprehensive RL training framework that jointly trains the problem composer and solver in an adversarial loop. **GAR** introduces an implicit curriculum learning mechanism, which aligns task difficulty with the prover's evolving capability. It thereby improves the training efficiency and enables stronger performance of proving advanced theorems. Experiments show that with **GAR** training, Goedel-Prover-V2-8B and DeepSeek-Prover-V2-7B achieve an average relative improvement in pass@32 of **4.20%** on MiniF2F-Test benchmark, while DeepSeek-Prover-V2's pass@32 on ProofNet-Test increases from 22.58% to **25.81%**. Beyond formal proving, **GAR** establishes a general RL paradigm for co-evolution of problem generation and solving under verifiable environments. The training code for this paper is open-sourced in https://github.com/RickySkywalker/GAR-Official