Online Learnability of Chain-of-Thought Verifiers: Soundness and Completeness Trade-offs
arXiv:2603.03538v1 Announce Type: new Abstract: Large language models with chain-of-thought generation have demonstrated great potential for producing complex mathematical proofs. However, their reasoning can often go astray, leading to increasing interest in formal and learned verifiers. A major challenge in learning verifiers, especially when their output will be used by the prover, is that this feedback loop may produce substantial distribution shift. Motivated by this challenge, we propose an online learning framework for learning chain-of-thought verifiers that, given a problem and a sequence of reasoning steps, check the correctness of the solution. Highlighting the asymmetric role of soundness (failure in catching errors in a proof) and completeness (flagging correct proofs as wrong) mistakes of the verifier, we introduce novel extensions of the Littlestone dimension which tightly characterize the mistake bounds for learning a verifier in the realizable setting. We provide opti
arXiv:2603.03538v1 Announce Type: new Abstract: Large language models with chain-of-thought generation have demonstrated great potential for producing complex mathematical proofs. However, their reasoning can often go astray, leading to increasing interest in formal and learned verifiers. A major challenge in learning verifiers, especially when their output will be used by the prover, is that this feedback loop may produce substantial distribution shift. Motivated by this challenge, we propose an online learning framework for learning chain-of-thought verifiers that, given a problem and a sequence of reasoning steps, check the correctness of the solution. Highlighting the asymmetric role of soundness (failure in catching errors in a proof) and completeness (flagging correct proofs as wrong) mistakes of the verifier, we introduce novel extensions of the Littlestone dimension which tightly characterize the mistake bounds for learning a verifier in the realizable setting. We provide optimal algorithms for finding the Pareto-frontier (the smallest total number of mistakes given a budget of soundness mistakes) as well as minimizing a linear combination of asymmetric costs. We further show how our learned verifiers can be used to boost the accuracy of a collection of weak provers, and enable generation of proofs beyond what they were trained on. With the mild assumption that one of the provers can generate the next reasoning step correctly with some minimal probability, we show how to learn a strong prover with small error and abstention rates.
Executive Summary
This article proposes an online learning framework for learning chain-of-thought verifiers, which are essential for ensuring the correctness of complex mathematical proofs generated by large language models. The framework addresses the challenge of substantial distribution shift in the feedback loop between the verifier and the prover. The authors introduce novel extensions of the Littlestone dimension to characterize the mistake bounds for learning a verifier in the realizable setting. They also provide optimal algorithms for finding the Pareto-frontier and minimizing asymmetric costs. The learned verifiers can boost the accuracy of weak provers and enable generation of proofs beyond their training data. This work has significant implications for the development of reliable and efficient proof verification systems in artificial intelligence and mathematics.
Key Points
- ▸ Proposes an online learning framework for learning chain-of-thought verifiers
- ▸ Introduces novel extensions of the Littlestone dimension for characterizing mistake bounds
- ▸ Provides optimal algorithms for finding the Pareto-frontier and minimizing asymmetric costs
Merits
Strength in Addressing Distribution Shift
The framework addresses the significant challenge of substantial distribution shift in the feedback loop between the verifier and the prover, which is essential for ensuring the reliability of proof verification systems.
Novel Extensions of the Littlestone Dimension
The authors introduce novel extensions of the Littlestone dimension, which provide a tight characterization of mistake bounds for learning a verifier in the realizable setting, and enable the development of more efficient and effective proof verification systems.
Demerits
Assumption of Minimal Probability of Correct Next Reasoning Step
The authors assume that one of the provers can generate the next reasoning step correctly with some minimal probability, which may not always be the case in practice, and may limit the applicability of the framework in certain scenarios.
Limited Evaluation of the Framework's Performance
The article does not provide a comprehensive evaluation of the framework's performance on real-world data, which may limit its practical applicability and adoption.
Expert Commentary
This article makes a significant contribution to the development of reliable and efficient proof verification systems, which is essential for ensuring the correctness and trustworthiness of complex mathematical proofs generated by large language models. The proposed framework addresses the challenge of substantial distribution shift in the feedback loop between the verifier and the prover, and provides a tight characterization of mistake bounds for learning a verifier in the realizable setting. However, the assumption of minimal probability of correct next reasoning step and the limited evaluation of the framework's performance may limit its practical applicability and adoption. Nevertheless, the article has significant implications for the development of artificial intelligence and mathematics, and can contribute to the advancement of these fields.
Recommendations
- ✓ Further evaluation of the framework's performance on real-world data is necessary to assess its practical applicability and adoption.
- ✓ The assumption of minimal probability of correct next reasoning step should be relaxed or addressed in future work to ensure the framework's applicability in a broader range of scenarios.