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Forward-Backward Reasoning in Large Language Models for Mathematical Verification

Paper · 2024 · Weisen Jiang, Han Shi, Longhui Yu et al.

Summary

The paper addresses a limitation of Self-Consistency in chain-of-thought math solving. Self-Consistency picks the answer that most sampled reasoning chains agree on, and its gains level off as more chains are sampled. Many of its errors are still recoverable, because some sampled chain often reaches the right answer even when the majority vote is wrong.

The authors add a verification step based on backward reasoning. They hide one number in the original question, attach a candidate answer, and ask the LLM to recover the hidden number. A candidate's backward score is the share of backward chains that reproduce the true masked value. FOBAR then ranks candidates by a weighted geometric mean of two scores: the forward vote share and this backward share.

The method is evaluated on six arithmetic benchmarks with three OpenAI models. FOBAR reaches state-of-the-art accuracy and beats both Self-Consistency and Self-Verification. The main lesson is that a very simple masking template, combined with forward voting, yields a more reliable answer selector. It also needs no extra training.

Key points

  • Data sets: AddSub, MultiArith, SingleEQ, SVAMP, GSM8K and AQuA.
  • Models: text-davinci-003, GPT-3.5-Turbo and GPT-4, prompted with CoT or ComplexCoT.
  • Number of chains: accuracy plateaus quickly as forward (M_F) and backward (M_B) chains increase. Only very small values (about ≤5 forward, ≤4 backward) noticeably hurt, so modest sampling budgets are enough.
  • Error analysis (GPT-3.5-Turbo, ComplexCoT):
    • About 60% of Self-Consistency failures (294 of 505) had at least one correct chain.
    • FOBAR fixed 54 of those 294 cases.
  • Extension and limits:
    • FOBAR also outperforms the baselines on two non-math tasks with GPT-3.5-Turbo: Date Understanding and Last Letter Concatenation.
    • Tasks like CSQA and StrategyQA remain open, because it is unclear which word to mask.

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 · China335 works · 43,758 citations🎓Hong Kong University of Science and TechnologyHong Kong · China183 works · 27,600 citations🏢Huawei TechnologiesShenzhen · China105 works · 15,705 citations🎓Southern University of Science and TechnologyShenzhen · China23 works · 1,381 citations

 Fields, topics and concepts

📚 Shelves 6The field of science this work belongs to (OpenAlex fields).Computer Science · 19,503 worksArtificial Intelligence · 18,805 worksArts and Humanities · 17,959 worksLanguage and Linguistics · 17,789 worksMathematics · 233 worksApplied Mathematics · 73 works

🧭 Topics 2Research subjects and languages the work is about.Language models & pretraining · 4,145 worksReasoning · 1,948 works

Cite

Cite as Weisen Jiang, Han Shi, Longhui Yu, Zhengying Liu, Yu Zhang, Zhenguo Li and James Kwok. 2024. Forward-Backward Reasoning in Large Language Models for Mathematical Verification. Findings of the Association for Computational Linguistics: ACL 2024.

The paper itself, not this page. BibTeX from the library's record of the work. DOI 10.18653/v1/2024.findings-acl.397 · arXiv 2308.07758

@inproceedings{jiang2024forward,
  title = {{Forward-Backward Reasoning in Large Language Models for Mathematical Verification}},
  author = {Weisen Jiang, Han Shi, Longhui Yu, Zhengying Liu, Yu Zhang, Zhenguo Li and James Kwok},
  booktitle = {Findings of the Association for Computational Linguistics: ACL 2024},
  year = {2024},
  doi = {10.18653/v1/2024.findings-acl.397},
  eprint = {2308.07758},
  archivePrefix = {arXiv},
  url = {https://doi.org/10.18653/v1/2024.findings-acl.397}
}

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