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arXiv · 2509.19702

Linear Transformers Implicitly Discover Unified Numerical Algorithms

Patrick Lutz·Aditya Gangrade·Hadi Daneshmand·Venkatesh Saligrama
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Abstract

We train a linear attention transformer on millions of masked-block matrix completion tasks: each prompt is masked low-rank matrix whose missing block may be (i) a scalar prediction target or (ii) an unseen kernel slice of Nystr\"om extrapolation. The model sees only input-output pairs and a mean-squared loss; it is given no normal equations, no handcrafted iterations, and no hint that the tasks are related. Surprisingly, after training, algebraic unrolling reveals the same parameter-free update rule across three distinct computational regimes (full visibility, rank-limited updates, and distributed computation). We prove that this rule achieves second-order convergence on full-batch problems, cuts distributed iteration complexity, and remains accurate with rank-limited attention. Thus, a transformer trained solely to patch missing blocks implicitly discovers a unified, resource-adaptive iterative solver spanning prediction, estimation, and Nystr\"om extrapolation, highlighting a powerful capability of in-context learning.

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Patrick Lutz, Aditya Gangrade, Hadi Daneshmand, Venkatesh Saligrama. 2025-09-24. Linear Transformers Implicitly Discover Unified Numerical Algorithms. https://arxiv.org/abs/2509.19702

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