OptProver revolutionizes formal theorem proving for optimization by combining expert iteration with utility-aware and perplexity-weighted preference optimization, achieving state-of-the-art results on new benchmarks while preserving general capabilities.
OptProver: Bridging Olympiad and Optimization through Continual Training in Formal Theorem Proving
Executive Impact: Revolutionizing Mathematical Verification
This paper introduces OptProver, a novel formal theorem proving model designed to bridge the gap between Olympiad-level problems and undergraduate optimization. It addresses critical challenges of distribution shift and catastrophic forgetting in continual training. OptProver achieves this through two key innovations: large-scale optimization-focused data curation via expert iteration, and a specialized preference learning objective that integrates perplexity-weighted optimization with a mechanism to penalize valid but non-progressing proof steps. Rigorous evaluation on OptBench, a new optimization benchmark, demonstrates OptProver's state-of-the-art Pass@1 and Pass@32 scores, solving over 55% of problems, while maintaining competitive performance on general theorem-proving tasks. This robust framework enables effective domain transfer without catastrophic forgetting, marking a significant advancement in automated theorem proving for optimization.
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OptBench Pass@32
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Problems Solved
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OptProver’s core methodology involves a multi-stage continual training framework. It starts with a strong Olympiad-level prover and uses large-scale self-distillation via expert iteration to generate high-quality proof trajectories. This process ensures the model learns from mathematically correct and discoverable proofs. To further refine its performance, OptProver employs two complementary preference optimization strategies: utility-aware preference optimization and perplexity-weighted DPO. These strategies are designed to penalize unhelpful tactics and stabilize training, enhancing both proving ability and search efficiency.
Key innovations include a novel benchmark called OptBench for undergraduate-level optimization problems. The utility-aware preference optimization method explicitly penalizes 'stagnant transitions' – syntactically valid but unhelpful tactics – guiding the model towards efficient proof trajectories. The perplexity-weighted DPO (PW-DPO) stabilizes training by modulating loss contributions based on token-level alignment with the reference distribution, preventing overfitting to low-probability data.
OptProver achieves state-of-the-art results on the OptBench benchmark, solving over 55% of problems. It significantly outperforms whole-proof generation models and standard step-level provers. Crucially, it demonstrates robust generalization, improving performance on general mathematical benchmarks like ProofNet without catastrophic forgetting, while maintaining parity on MiniF2F. This validates its ability to transfer domain-specific knowledge without sacrificing broader reasoning capabilities.
55.25%
State-of-the-Art Pass@32 on OptBench
OptProver demonstrates superior performance on optimization tasks, significantly outperforming previous models and solving over half of the benchmark problems.
OptProver's Continual Training Pipeline
Olympiad-level Base Model
→
Large-Scale Self-Distillation (Expert Iteration)
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Optimization-Focused Data Curation (OptLib, DeepTheorem)
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Utility-Aware Preference Optimization
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Perplexity-Weighted DPO
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OptProver: Domain-Adapted, Robust Prover
| Method | Basic | Convex | Algorithmic | Avg |
|---|---|---|---|---|
| DeepSeek-Prover-V2-7B | 19.01% | 31.85% | 6.25% | 18.75% |
| BFS-Prover-V2-7B | 36.36% | 44.44% | 16.67% | 32.00% |
| OptProver (EI + PW-UAPO) | 55.37% | 62.22% | 48.61% | 55.25% |
The table highlights OptProver's significant performance improvements across all optimization subcategories compared to state-of-the-art baselines. |
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Real-World Application: ADMM Convergence Proofs
One of OptProver's notable achievements is its ability to handle complex inequalities found in the convergence analysis of the Alternating Direction Method of Multipliers (ADMM). Traditional provers struggle with these proofs due to the required deep understanding of mathematical analysis and optimization-specific definitions. OptProver, through its specialized training, successfully navigates these challenges, demonstrating its capacity for advanced theoretical verification in applied mathematics.
Outcome: Automated verification of ADMM convergence properties, previously a manual and error-prone process, significantly accelerating research and development in optimization algorithms.
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Your AI Prover Implementation Roadmap
A structured approach to integrating OptProver into your existing mathematical verification workflows.
Phase 1: Foundation Building
Establish Lean 4 environment and integrate OptLib for core optimization definitions.
Phase 2: Data Curation & Pre-training
Autoformalize textbooks (Convex/Real Analysis) and generate initial expert iteration traces.
Phase 3: Iterative Refinement
Apply utility-aware and perplexity-weighted DPO on self-generated proofs, focusing on efficiency.
Phase 4: Validation & Deployment
Evaluate on OptBench, MiniF2F, and ProofNet, ensuring robustness and domain transfer.
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