OptProver: Bridging Olympiad and Optimization through Continual Training in Formal Theorem Proving

2026-04-26

AI Summary

Current formal theorem provers excel at Olympiad-level mathematics but suffer performance degradation on undergraduate optimization problems—such as those involving convexity and optimality conditions—due to distributional shift. This work proposes a domain-specific transfer approach for optimization: leveraging a strong Olympiad-level prover, it constructs a large-scale optimization dataset via expert iteration and introduces a novel preference learning objective that combines perplexity weighting with penalties for non-progress steps to guide efficient proof search. Evaluated on a newly curated optimization benchmark, the method achieves state-of-the-art Pass@1 and Pass@32 performance among models of comparable scale, while maintaining competitive results on general theorem-proving tasks, thereby enabling effective cross-domain transfer without catastrophic forgetting.

Technology Category

• Search and Optimization: Learning to Search
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Optimization

Application Category

• Search and Retrieval-Augmented AI: Vertical and domain-specific search
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

Recent advances in formal theorem proving have focused on Olympiad-level mathematics, leaving undergraduate domains largely unexplored. Optimization, fundamental to machine learning, operations research, and scientific computing, remains underserved by existing provers. Its reliance on domain-specific formalisms (convexity, optimality conditions, and algorithmic analysis) creates significant distribution shift, making naive domain transfer ineffective. We present OptProver, a trained model that achieves robust transfer from Olympiad to undergraduate optimization. Starting from a strong Olympiad-level prover, our pipeline mitigates distribution shift through two key innovations. First, we employ large-scale optimization-focused data curation via expert iteration. Second, we introduce a specialized preference learning objective that integrates perplexity-weighted optimization with a mechanism to penalize valid but non-progressing proof steps. This not only addresses distribution shifts but also guides the search toward efficient trajectories. To enable rigorous evaluation, we construct a novel benchmark in Lean 4 focused on optimization. On this benchmark, OptProver achieves state-of-the-art Pass@1 and Pass@32 among comparably sized models while maintaining competitive performance on general theorem-proving tasks, demonstrating effective domain transfer without catastrophic forgetting.

Problem

• formal theorem proving
• optimization
• distribution shift
• domain transfer
• undergraduate mathematics

Innovation

• continual training
• distribution shift mitigation
• preference learning
• formal theorem proving
• optimization benchmark