arxiv: 2604.23712 · v2 · submitted 2026-04-26 · 💻 cs.LG · cs.AI

Recognition: unknown

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

Chenyi Li , Yanchen Nie , Zhenyu Ming , Gong Zhang , Kun Yuan , Zaiwen Wen

Authors on Pith no claims yet

Pith reviewed 2026-05-08 06:30 UTC · model grok-4.3

classification 💻 cs.LG cs.AI

keywords formal theorem provingoptimizationdomain adaptationpreference learningLean 4expert iterationcontinual training

0 comments

The pith

OptProver adapts an Olympiad-level prover to undergraduate optimization tasks through targeted data curation and a new preference objective.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper addresses the gap where formal theorem provers handle Olympiad math well but struggle with optimization problems common in machine learning and operations research. It starts from an existing strong Olympiad prover and applies expert iteration to build large amounts of optimization-specific training data. A specialized preference learning objective then weights steps by perplexity while penalizing valid steps that fail to advance the proof. On a new Lean 4 benchmark focused on optimization, the resulting model reaches top Pass@1 and Pass@32 scores among models of similar size and keeps competitive results on general theorem proving. A reader would care because this demonstrates a practical path to extend formal verification into applied domains without erasing prior knowledge.

Core claim

Starting from a strong Olympiad-level prover, OptProver mitigates distribution shift through 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. On a novel Lean 4 benchmark for optimization, it achieves state-of-the-art Pass@1 and Pass@32 among comparably sized models while maintaining competitive performance on general theorem-proving tasks.

What carries the argument

The central mechanism is expert iteration for generating optimization-focused training data paired with a perplexity-weighted preference objective that penalizes non-progressing proof steps.

Load-bearing premise

The combination of expert iteration data curation and the perplexity-weighted preference objective with penalties for non-progressing steps is sufficient to reduce the distribution shift between Olympiad and optimization domains.

What would settle it

If OptProver fails to exceed the Pass@1 scores of other comparably sized models on the new optimization benchmark or shows a large drop in performance on general theorem-proving tasks, the claim of effective transfer without forgetting would be refuted.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

OptProver adds a Lean 4 optimization benchmark and a perplexity-weighted preference objective for domain transfer from olympiad proving, but the abstract gives no ablations or numbers so the objective's contribution stays unproven.

read the letter

The main point is that this paper builds a new Lean 4 benchmark for undergraduate optimization problems and trains a model that starts from an olympiad prover, using expert iteration for data plus a custom preference loss. It claims state-of-the-art pass rates on the new set while keeping performance on general tasks. That combination is the core offering. The benchmark itself is new and targets a real gap—optimization proofs involve convexity, optimality conditions, and algorithmic reasoning that standard math datasets skip. The preference objective weights by perplexity and adds a penalty for valid steps that do not advance the proof; the idea is to reduce distribution shift and steer search toward useful trajectories. Those two pieces together are what the authors present as their advance over plain continual training. The work does a service by naming the domain shift problem explicitly and supplying an evaluation set that others can use. The no-forgetting result, if it holds, would also be useful for anyone doing continual learning in theorem provers. The soft spots sit in the experimental claims. The abstract states SOTA Pass@1 and Pass@32 numbers but supplies no baselines, no statistical tests, no ablation tables, and no description of benchmark size or construction details. Without those, it is impossible to tell whether the specialized objective adds signal beyond the volume of curated data. The stress-test concern lands: expert iteration is a standard tool, so the transfer success could be explained by data scale alone. If the full paper contains controlled comparisons, that would change the picture; on the current evidence the central methodological claim rests on unisolated factors. The citation pattern looks light on prior optimization formalization work, which is consistent with the claim that the area is underserved but also means the novelty of the benchmark needs careful checking. This paper is for researchers in formal theorem proving and machine learning for mathematics who want to move beyond olympiad problems into applied domains. A reader who needs a Lean optimization test set or ideas for domain-specific objectives will get something concrete from it. It deserves a serious referee because the benchmark is a concrete resource and the transfer-without-forgetting angle is worth testing, even if the current write-up requires substantial additions on experiments and ablations before the claims can be evaluated. I would send it to review with a request for those details.

Referee Report

3 major / 2 minor

Summary. The paper introduces OptProver, which starts from an Olympiad-level formal theorem prover and applies continual training to address distribution shift into undergraduate optimization. It uses two innovations: large-scale optimization data curation via expert iteration, and a perplexity-weighted preference objective that penalizes valid but non-progressing proof steps. A new Lean 4 benchmark for optimization is constructed, on which OptProver reports SOTA Pass@1 and Pass@32 among comparably sized models while remaining competitive on general theorem-proving tasks and avoiding catastrophic forgetting.

Significance. If the empirical claims hold after proper validation, the work would be significant for demonstrating robust domain transfer in formal theorem proving to an important but underserved area (optimization), with potential value for ML, operations research, and scientific computing. The new benchmark could serve as a useful resource, and the combination of expert iteration with a specialized preference objective offers a concrete approach to mitigating forgetting in continual learning settings for provers.

major comments (3)

[Experimental results] The abstract and experimental claims assert SOTA Pass@1/Pass@32 results and no forgetting, yet the manuscript provides no experimental details, baselines, statistical significance, number of runs, or ablation studies. This is load-bearing for the central claim of effective domain transfer.
[Methods / Preference learning objective] No controlled comparison isolates the contribution of the perplexity-weighted preference objective (with penalization of non-progressing steps) from the expert-iteration data curation alone. Without this, the SOTA numbers on the new benchmark could be explained by data volume rather than the claimed methodological advance in reducing distribution shift.
[Benchmark construction] The construction of the novel optimization benchmark (problem selection criteria, coverage of convexity/optimality conditions/algorithmic analysis, verification process, and quantification of distribution shift from Olympiad data) lacks sufficient detail for reproducibility or for assessing whether the transfer results are robust.

minor comments (2)

[Abstract] The abstract refers to 'large-scale' data curation without specifying scale, sources, or filtering criteria.
[Methods] Notation and equations for the perplexity-weighted objective and penalization mechanism should be presented explicitly in the main text rather than described at high level.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed feedback. We address each major comment below and commit to revising the manuscript to provide the requested clarifications and additional experiments where feasible.

read point-by-point responses

Referee: [Experimental results] The abstract and experimental claims assert SOTA Pass@1/Pass@32 results and no forgetting, yet the manuscript provides no experimental details, baselines, statistical significance, number of runs, or ablation studies. This is load-bearing for the central claim of effective domain transfer.

Authors: We agree that the current version lacks sufficient experimental detail to fully substantiate the claims. In the revised manuscript we will add a dedicated experimental section reporting all baselines, the number of independent runs, statistical significance tests, variance across runs, and ablation studies. These additions will directly support the reported Pass@1/Pass@32 results and the claim of no catastrophic forgetting. revision: yes
Referee: [Methods / Preference learning objective] No controlled comparison isolates the contribution of the perplexity-weighted preference objective (with penalization of non-progressing steps) from the expert-iteration data curation alone. Without this, the SOTA numbers on the new benchmark could be explained by data volume rather than the claimed methodological advance in reducing distribution shift.

Authors: We acknowledge that the present manuscript does not contain a controlled ablation separating the preference objective from expert-iteration data curation. We will add such an ablation in the revision, training a variant using only expert iteration and comparing it against the full OptProver pipeline. This will demonstrate the incremental benefit of the perplexity-weighted preference term in guiding efficient proof trajectories beyond data volume alone. revision: yes
Referee: [Benchmark construction] The construction of the novel optimization benchmark (problem selection criteria, coverage of convexity/optimality conditions/algorithmic analysis, verification process, and quantification of distribution shift from Olympiad data) lacks sufficient detail for reproducibility or for assessing whether the transfer results are robust.

Authors: We will expand the benchmark section with explicit details on problem selection criteria, the distribution of topics (convexity, optimality conditions, algorithmic analysis), the Lean 4 verification pipeline, and quantitative measures of distribution shift (e.g., perplexity or embedding distance) relative to Olympiad data. These additions will improve reproducibility and allow readers to evaluate the robustness of the reported transfer. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical claims rest on new benchmark and standard training techniques

full rationale

The paper describes an empirical pipeline of expert iteration for data curation plus a perplexity-weighted preference objective, followed by evaluation on a newly constructed Lean 4 optimization benchmark. No equations, self-definitions, or self-citations reduce the reported Pass@1/Pass@32 scores or domain-transfer claims to quantities defined by the same inputs. The performance numbers are measured outcomes on held-out data, not identities or fitted renamings. The central claim of effective transfer without forgetting is presented as an experimental result, not a mathematical tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, axioms, or invented entities; the approach relies on standard machine-learning practices whose details are not stated.

pith-pipeline@v0.9.0 · 5521 in / 1076 out tokens · 25806 ms · 2026-05-08T06:30:35.997514+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

35 extracted references · 13 canonical work pages · 1 internal anchor

[1]

Large language models for mathematical reasoning: Progresses and challenges.arXiv preprint arXiv:2402.00157, 2024

Ahn, J., Verma, R., Lou, R., Liu, D., Zhang, R., and Yin, W. Large language models for mathematical reasoning: Progresses and challenges.arXiv preprint arXiv:2402.00157, 2024

work page arXiv 2024
[2]

G., Rowland, M., Piot, B., Guo, D., Calandriello, D., Valko, M., and Munos, R

Azar, M. G., Rowland, M., Piot, B., Guo, D., Calandriello, D., Valko, M., and Munos, R. A general theoretical paradigm to understand learning from human preferences, 2023

2023
[3]

H., Qazi, I

Azeemi, A. H., Qazi, I. A., and Raza, A. A. Language model-driven data pruning enables efficient active learning. arXiv preprint arXiv:2410.04275, 2024

work page arXiv 2024
[4]

Bansal, A.; Loos, S.; Rabe, M.; Szegedy, C.; and Wilcox, S

Azerbayev, Z., Piotrowski, B., Schoelkopf, H., Ayers, E. W., Radev, D., and Avigad, J. Proofnet: Autoformalizing and formally proving undergraduate-level mathematics.arXiv preprint arXiv:2302.12433, 2023

work page arXiv 2023
[5]

arXiv preprint arXiv:2307.08701 , year=

Chen, L., Li, S., Yan, J., Wang, H., Gunaratna, K., Yadav, V., Tang, Z., Srinivasan, V., Zhou, T., Huang, H., et al. Alpagasus: Training a better alpaca with fewer data.arXiv preprint arXiv:2307.08701, 2023

work page arXiv 2023
[6]

F., Leike, J., Brown, T., Martic, M., Legg, S., and Amodei, D

Christiano, P. F., Leike, J., Brown, T., Martic, M., Legg, S., and Amodei, D. Deep reinforcement learning from human preferences.Advances in neural information processing systems, 30, 2017

2017
[7]

The Lean theorem prover (system description)

De Moura, L., Kong, S., Avigad, J., Van Doorn, F., and von Raumer, J. The Lean theorem prover (system description). InInternational Conference on Automated Deduction, pp. 378–388. Springer, 2015

2015
[8]

Olympiadbench: A challenging benchmark for promoting AGI with Olympiad-level bilingual multimodal scientific problems

He, C., Luo, R., Bai, Y., Hu, S., Thai, Z., Shen, J., Hu, J., Han, X., Huang, Y., Zhang, Y., et al. Olympiadbench: A challenging benchmark for promoting AGI with Olympiad-level bilingual multimodal scientific problems. In Proceedings of the 62nd Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pp. 3828–3850, 2024

2024
[9]

Measuring Mathematical Problem Solving With the MATH Dataset

Hendrycks, D., Burns, C., Kadavath, S., Arora, A., Basart, S., Tang, E., Song, D., and Steinhardt, J. Measuring Mathematical Problem Solving With the MATH Dataset. In Vanschoren, J. and Yeung, S. (eds.),Proceedings of the Neural Information Processing Systems Track on Datasets and Benchmarks, volume 1, 2021

2021
[10]

A., Milan, K., Quan, J., Ramalho, T., Grabska-Barwinska, A., et al

Kirkpatrick, J., Pascanu, R., Rabinowitz, N., Veness, J., Desjardins, G., Rusu, A. A., Milan, K., Quan, J., Ramalho, T., Grabska-Barwinska, A., et al. Overcoming catastrophic forgetting in neural networks.Proceedings of the national academy of sciences, 114(13):3521–3526, 2017

2017
[11]

Hypertree proof search for neural theorem proving.Advances in neural information processing systems, 35: 26337–26349, 2022

Lample, G., Lacroix, T., Lachaux, M.-A., Rodriguez, A., Hayat, A., Lavril, T., Ebner, G., and Martinet, X. Hypertree proof search for neural theorem proving.Advances in neural information processing systems, 35: 26337–26349, 2022

2022
[12]

Solving Quantitative Reasoning Problems with Language Models

Lewkowycz, A., Andreassen, A., Dohan, D., Dyer, E., Michalewski, H., Ramasesh, V., Slone, A., Anil, C., Schlag, I., Gutman-Solo, T., Wu, Y., Neyshabur, B., Gur-Ari, G., and Misra, V. Solving Quantitative Reasoning Problems with Language Models. In Koyejo, S., Mohamed, S., Agarwal, A., Belgrave, D., Cho, K., and Oh, A. (eds.), Advances in Neural Informatio...

2022
[13]

Formalization of Convergence Rates of Four First-order Algorithms for Convex Optimization.Journal of Automated Reasoning, 69(4):28, 2025

Li, C., Wang, Z., He, W., Wu, Y., Xu, S., and Wen, Z. Formalization of Convergence Rates of Four First-order Algorithms for Convex Optimization.Journal of Automated Reasoning, 69(4):28, 2025

2025
[14]

Formalization of algorithms for optimization with block structures.Science China Mathematics, pp

Li, C., Wang, Z., Bai, Y., Duan, Y., Gao, Y., Hao, P., and Wen, Z. Formalization of algorithms for optimization with block structures.Science China Mathematics, pp. 1–18, 2026

2026
[15]

Goedel-Prover-V2: Scaling Formal Theorem Proving with Scaffolded Data Synthesis and Self-Correction

Lin, Y., Tang, S., Lyu, B., Yang, Z., Chung, J.-H., Zhao, H., Jiang, L., Geng, Y., Ge, J., Sun, J., et al. Goedel- prover-v2: Scaling formal theorem proving with scaffolded data synthesis and self-correction.arXiv preprint arXiv:2508.03613, 2025

work page arXiv 2025
[16]

Not all tokens are what you need for pretraining.Advances in Neural Information Processing Systems, 37:29029–29063, 2024

Lin, Z., Gou, Z., Gong, Y., Liu, X., Xu, R., Lin, C., Yang, Y., Jiao, J., Duan, N., Chen, W., et al. Not all tokens are what you need for pretraining.Advances in Neural Information Processing Systems, 37:29029–29063, 2024

2024
[17]

SimPO: Simple Preference Optimization with a Reference-Free Reward, 2024

Meng, Y., Xia, M., and Chen, D. SimPO: Simple Preference Optimization with a Reference-Free Reward, 2024

2024
[18]

Moore, R. C. and Lewis, W. Intelligent selection of language model training data. InProceedings of the ACL 2010 conference short papers, pp. 220–224, 2010

2010
[19]

Moura, L. d. and Ullrich, S. The Lean 4 theorem prover and programming language. InInternational Conference on Automated Deduction, pp. 625–635. Springer, 2021. 12

2021
[20]

Training language models to follow instructions with human feedback.Advances in neural information processing systems, 35:27730–27744, 2022

Ouyang, L., Wu, J., Jiang, X., Almeida, D., Wainwright, C., Mishkin, P., Zhang, C., Agarwal, S., Slama, K., Ray, A., et al. Training language models to follow instructions with human feedback.Advances in neural information processing systems, 35:27730–27744, 2022

2022
[21]

Poole, D

Polu, S. and Sutskever, I. Generative language modeling for automated theorem proving.arXiv preprint arXiv:2009.03393, 2020

work page arXiv 2009
[22]

D., Ermon, S., and Finn, C

Rafailov, R., Sharma, A., Mitchell, E., Manning, C. D., Ermon, S., and Finn, C. Direct preference optimization: Your language model is secretly a reward model.Advances in neural information processing systems, 36:53728– 53741, 2023

2023
[23]

DeepSeek-Prover-V2: Advancing Formal Mathematical Reasoning via Reinforcement Learning for Subgoal Decomposition

Ren, Z., Shao, Z., Song, J., Xin, H., Wang, H., Zhao, W., Zhang, L., Fu, Z., Zhu, Q., Yang, D., et al. Deepseek- prover-v2: Advancing formal mathematical reasoning via reinforcement learning for subgoal decomposition.arXiv preprint arXiv:2504.21801, 2025

work page internal anchor Pith review arXiv 2025
[24]

Princeton Landmarks in Mathematics and Physics

Rockafellar, R.Convex Analysis. Princeton Landmarks in Mathematics and Physics. Princeton University Press,
[25]

REAL-Prover: Retrieval Augmented Lean Prover for Mathematical Reasoning, 2025

Shen, Z., Huang, N., Yang, F., Wang, Y., et al. REAL-Prover: Retrieval Augmented Lean Prover for Mathematical Reasoning, 2025

2025
[26]

Springer, 2006

Tao, T.Analysis I, volume 1. Springer, 2006

2006
[27]

Kimina-Prover Preview: Towards Large Formal Reasoning Models with Reinforcement Learning

Wang, H., Unsal, M., Lin, X., Baksys, M., Liu, J., Santos, M. D., Sung, F., Vinyes, M., Ying, Z., Zhu, Z., et al. Kimina-prover preview: Towards large formal reasoning models with reinforcement learning.arXiv preprint arXiv:2504.11354, 2025

work page arXiv 2025
[28]

R., Lin, C.-Y., Chen, Y.-N., Sun, S.-H., and Lee, H.-y

Wu, C.-C., Tam, Z. R., Lin, C.-Y., Chen, Y.-N., Sun, S.-H., and Lee, H.-y. Mitigating Forgetting in LLM Fine-Tuning via Low-Perplexity Token Learning.arXiv preprint arXiv:2501.14315, 2025

work page arXiv 2025
[29]

Internlm2

Wu, Z., Huang, S., Zhou, Z., Ying, H., Wang, J., Lin, D., and Chen, K. Internlm2.5-stepprover: Advancing automated theorem proving via expert iteration on large-scale lean problems.arXiv preprint arXiv:2410.15700, 2024

work page arXiv 2024
[30]

M., Santurkar, S., Ma, T., and Liang, P

Xie, S. M., Santurkar, S., Ma, T., and Liang, P. S. Data selection for language models via importance resampling. Advances in Neural Information Processing Systems, 36:34201–34227, 2023

2023
[31]

Bfs-prover: Scalable best-first tree search for LLM-based automatic theorem proving

Xin, R., Xi, C., Yang, J., Chen, F., Wu, H., Xiao, X., Sun, Y., Zheng, S., and Ding, M. Bfs-prover: Scalable best-first tree search for LLM-based automatic theorem proving. InProceedings of the 63rd Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pp. 32588–32599, 2025

2025
[32]

Scaling up multi-turn off-policy rl and multi-agent tree search for llm step-provers.arXiv preprint arXiv:2509.06493, 2025b

Xin, R., Zheng, Z., Nie, Y., Yuan, K., and Xiao, X. Scaling up multi-turn off-policy RL and multi-agent tree search for llm step-provers.arXiv preprint arXiv:2509.06493, 2025

work page arXiv 2025
[33]

J., and Anandkumar, A

Yang, K., Swope, A., Gu, A., Chalamala, R., Song, P., Yu, S., Godil, S., Prenger, R. J., and Anandkumar, A. Leandojo: Theorem proving with retrieval-augmented language models.Advances in Neural Information Processing Systems, 36:21573–21612, 2023

2023
[34]

Deeptheorem: Advancing LLM reasoning for theorem proving through natural language and reinforcement learning.arXiv preprint arXiv:2505.23754, 2025

Zhang, Z., Xu, J., He, Z., Liang, T., Liu, Q., Li, Y., Song, L., Liang, Z., Zhang, Z., Wang, R., et al. Deeptheorem: Advancing LLM reasoning for theorem proving through natural language and reinforcement learning.arXiv preprint arXiv:2505.23754, 2025

work page arXiv 2025
[35]

MiniF2F: a cross-system benchmark for formal Olympiad-level mathematics

Zheng, K., Han, J. M., and Polu, S. Minif2f: a cross-system benchmark for formal Olympiad-level mathematics. arXiv preprint arXiv:2109.00110, 2021. 13 A Brief Introduction to Lean Lean is a general-purpose functional programming language and interactive theorem prover based on dependent type theory. It is primarily designed for formal verification. Mathem...

work page arXiv 2021