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| Autori principali: | , , , , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2510.00184 |
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| _version_ | 1866908571049918464 |
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| author | Bai, Xiaoyan Pres, Itamar Deng, Yuntian Tan, Chenhao Shieber, Stuart Viégas, Fernanda Wattenberg, Martin Lee, Andrew |
| author_facet | Bai, Xiaoyan Pres, Itamar Deng, Yuntian Tan, Chenhao Shieber, Stuart Viégas, Fernanda Wattenberg, Martin Lee, Andrew |
| contents | Language models are increasingly capable, yet still fail at a seemingly simple task of multi-digit multiplication. In this work, we study why, by reverse-engineering a model that successfully learns multiplication via \emph{implicit chain-of-thought}, and report three findings: (1) Evidence of long-range structure: Logit attributions and linear probes indicate that the model encodes the necessary long-range dependencies for multi-digit multiplication. (2) Mechanism: the model encodes long-range dependencies using attention to construct a directed acyclic graph to ``cache'' and ``retrieve'' pairwise partial products. (3) Geometry: the model implements partial products in attention heads by forming Minkowski sums between pairs of digits, and digits are represented using a Fourier basis, both of which are intuitive and efficient representations that the standard fine-tuning model lacks. With these insights, we revisit the learning dynamics of standard fine-tuning and find that the model converges to a local optimum that lacks the required long-range dependencies. We further validate this understanding by introducing an auxiliary loss that predicts the ``running sum'' via a linear regression probe, which provides an inductive bias that enables the model to successfully learn multi-digit multiplication. In summary, by reverse-engineering the mechanisms of an implicit chain-of-thought model we uncover a pitfall for learning long-range dependencies in Transformers and provide an example of how the correct inductive bias can address this issue. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_00184 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Why Can't Transformers Learn Multiplication? Reverse-Engineering Reveals Long-Range Dependency Pitfalls Bai, Xiaoyan Pres, Itamar Deng, Yuntian Tan, Chenhao Shieber, Stuart Viégas, Fernanda Wattenberg, Martin Lee, Andrew Machine Learning Artificial Intelligence Language models are increasingly capable, yet still fail at a seemingly simple task of multi-digit multiplication. In this work, we study why, by reverse-engineering a model that successfully learns multiplication via \emph{implicit chain-of-thought}, and report three findings: (1) Evidence of long-range structure: Logit attributions and linear probes indicate that the model encodes the necessary long-range dependencies for multi-digit multiplication. (2) Mechanism: the model encodes long-range dependencies using attention to construct a directed acyclic graph to ``cache'' and ``retrieve'' pairwise partial products. (3) Geometry: the model implements partial products in attention heads by forming Minkowski sums between pairs of digits, and digits are represented using a Fourier basis, both of which are intuitive and efficient representations that the standard fine-tuning model lacks. With these insights, we revisit the learning dynamics of standard fine-tuning and find that the model converges to a local optimum that lacks the required long-range dependencies. We further validate this understanding by introducing an auxiliary loss that predicts the ``running sum'' via a linear regression probe, which provides an inductive bias that enables the model to successfully learn multi-digit multiplication. In summary, by reverse-engineering the mechanisms of an implicit chain-of-thought model we uncover a pitfall for learning long-range dependencies in Transformers and provide an example of how the correct inductive bias can address this issue. |
| title | Why Can't Transformers Learn Multiplication? Reverse-Engineering Reveals Long-Range Dependency Pitfalls |
| topic | Machine Learning Artificial Intelligence |
| url | https://arxiv.org/abs/2510.00184 |