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FoE: Forest of Errors Makes the First Solution the Best in Large Reasoning Models

Paper · 2026 · Kehan Jiang, Haonan Dong, Zhaolu Kang et al.

Summary

Large Reasoning Models such as DeepSeek-R1 often produce several alternative solutions inside a single chain of thought. This makes outputs long and costly, and the authors ask whether the extra exploration actually helps. Comparing the first solution (First) with later ones (Subs), they find that the first attempt is usually the most reliable. Later attempts rarely fix a wrong first answer and sometimes overturn a correct one. To explain this, they represent reasoning errors as a branching structure called the Forest of Errors (FoE). In this structure, root mistakes spawn dependent child errors, and error generation tracks token entropy and its variance. They support the analysis with a probabilistic branching-process argument. Building on this, they propose RED, a training-free method with two parts. The first uses entropy-triggered intervention to strengthen First. The second stops decoding early once a dual-consistency check confirms the answer. Across six backbones and five benchmarks, RED improves accuracy while cutting token use substantially. This suggests that longer test-time reasoning can add errors rather than remove them.

Key points

  • Error analysis: First is the better solution in up to 93.7% of cases.
    • When First is wrong, Subs stay wrong 75.4%–82.8% of the time and fix it in at most about 7.1% of cases.
    • Subs turn a correct First into a wrong answer in up to 21.2% of cases.
  • Models and benchmarks: Backbones were Qwen3-thinking (8B, 32B), DeepSeek-R1-Distill-Qwen (7B, 32B) and DeepSeek-R1-Distill-Llama (8B, 70B). Benchmarks were GSM8K, MATH500, AIME 2024, AIME 2025 and GPQA-Diamond.
  • How RED works:
    • It watches entropy over a sliding window (L=15, threshold 2.4) and injects a corrective prompt when the trigger fires.
    • It probes every 2 steps using 4 prompt templates with 12 samples each.
    • It exits early only when answers within each template agree at least 60% of the time and the dominant answers match across all templates.
  • Main results: Against eight baselines, including DEER, AlphaOne, DAST and S-GRPO, RED gained 0.3–5.6 points (3.2%–19.0%) and used 37.7%–70.4% fewer tokens.
  • Error reduction and cost: On AIME25, RED lowered the FoE metrics by 41.0%–68.0%. Its worst-case extra latency averages only 4.6%.

Our summary of the paper, in our own words — not its text. The work belongs to its authors; read it at the links above.

 Institutions

🎓Peking UniversityBeijing · China244 works · 39,359 citations

 Fields, topics and concepts

📚 Shelves 5The field of science this work belongs to (OpenAlex fields).Computer Science · 16,667 worksArtificial Intelligence · 16,194 worksArts and Humanities · 16,039 worksLanguage and Linguistics · 15,891 worksComputer Vision and Pattern Recognition · 2,869 works

🧭 Topics 1Research subjects and languages the work is about.Reasoning · 1,426 works

Cite

Cite as Kehan Jiang, Haonan Dong, Zhaolu Kang, Zhengzhou Zhu and Guojie Song. 2026. FoE: Forest of Errors Makes the First Solution the Best in Large Reasoning Models. Proceedings of the 64th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers).

The paper itself, not this page. BibTeX from the library's record of the work. DOI 10.18653/v1/2026.acl-long.1128

@inproceedings{jiang2026foe,
  title = {{FoE: Forest of Errors Makes the First Solution the Best in Large Reasoning Models}},
  author = {Kehan Jiang, Haonan Dong, Zhaolu Kang, Zhengzhou Zhu and Guojie Song},
  booktitle = {Proceedings of the 64th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)},
  year = {2026},
  doi = {10.18653/v1/2026.acl-long.1128},
  url = {https://doi.org/10.18653/v1/2026.acl-long.1128}
}

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