Why Can't Transformers Learn Multiplication? Reverse-Engineering Implicit Chain-of-Thought Reveals Challenges of Learning Long-Range Dependencies

ICLR 2026 Conference Submission13243 Authors

Keywords: mechanistic interpretability, arithmetic reasoning, feature geometry

TL;DR: We study why transformers can't learn multi-digit multiplication by reverse-engineering a model that successfully learns multiplication via process supervision called implicit chain-of-thought.

Abstract:

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.

Primary Area: interpretability and explainable AI

Submission Number: 13243