通过多重验证器并行检测错误，提升数学解题的准确性与效率。
Pessimistic Verification for Open Ended Math Questions
- 采用并行验证器，任一发现错误即拒绝解法，增强纠错能力。
- 在IMO 2025和MathArena Apex 2025上显著提升解题准确率与效率。
- 适合高难度数学竞赛问题求解，尤其适用于强模型验证场景。
自动验证是构建数学求解智能体和强化学习系统的关键组件，但常面临泛化性差、性能不足和成本效率低的问题。我们发现验证能力受限的核心在于错误检测能力，提出悲观验证范式：若多个并行验证器中任意一个识别出缺陷，则拒绝该解法。进一步提出渐进式悲观验证，通过细粒度证明分解显著提升验证的准确性和效率。本方法在性能和令牌效率上均优于扩展长思维链（long CoT）及主流验证流程。值得注意的是，分析表明现有基准因固有标注误差低估了该方法在更强模型上的表现。为验证有效性，我们在IMO 2025和MathArena Apex 2025数据集上应用基于验证的求解流程，结果显示，采用渐进式悲观验证的流程在使用前沿模型时，对高难度竞赛级数学问题实现了显著的准确率与效率提升。代码已开源：https://github.com/THUNLP-MT/pverify。
原文摘要 · Abstract (English)
Automatic verification is a critical component in building math-solving agents and reinforcement learning, yet it often falls short in generalizability, performance, and cost-efficiency. Identifying that the primary bottleneck of verification lies in error detection capability, we propose pessimistic verification, a paradigm of agentic workflows that rejects a solution if any of multiple parallel verifiers identifies a flaw. We further introduce progressive pessimistic verification, which employs fine-grained proof decomposition to significantly enhance verification accuracy and efficiency. Our approach surpasses the performance and token efficiency of extended long chain-of-thought (long CoT) and mainstream verification workflows, crucially, our analysis reveals that existing benchmarks underestimate its effectiveness on stronger models due to inherent annotation errors. To further validate the effectiveness of our method, we applied a verification-based solving workflow on the IMO 2025 and MathArena Apex 2025 datasets, where the workflow with progressive pessimistic verification exhibits remarkable improvements in both efficiency and accuracy on highly challenging contest-level math problems with state-of-the-art models. Code is available at https://github.com/THUNLP-MT/pverify.
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