Transformers Learn Higher-Order Optimization Methods for In-Context Learning: A Study with Linear Models

• URL: http://arxiv.org/abs/2310.17086v2
• Date: Fri, 31 May 2024 20:37:54 GMT
• Title: Transformers Learn Higher-Order Optimization Methods for In-Context Learning: A Study with Linear Models
• Authors: Deqing Fu, Tian-Qi Chen, Robin Jia, Vatsal Sharan,
• Abstract summary: We show that Transformers learn to approximate higher-order optimization methods for in-context linear regression. For in-context linear regression, Transformers share a similar convergence rate as Iterative Newton's Method. We also show that Transformers can learn in-context on ill-conditioned data, a setting where Gradient Descent struggles but Iterative Newton succeeds.
• Score: 23.944430707096103
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Transformers excel at in-context learning (ICL) -- learning from demonstrations without parameter updates -- but how they do so remains a mystery. Recent work suggests that Transformers may internally run Gradient Descent (GD), a first-order optimization method, to perform ICL. In this paper, we instead demonstrate that Transformers learn to approximate higher-order optimization methods for ICL. For in-context linear regression, Transformers share a similar convergence rate as Iterative Newton's Method; both are exponentially faster than GD. Empirically, predictions from successive Transformer layers closely match different iterations of Newton's Method linearly, with each middle layer roughly computing 3 iterations; thus, Transformers and Newton's method converge at roughly the same rate. In contrast, Gradient Descent converges exponentially more slowly. We also show that Transformers can learn in-context on ill-conditioned data, a setting where Gradient Descent struggles but Iterative Newton succeeds. Finally, to corroborate our empirical findings, we prove that Transformers can implement $k$ iterations of Newton's method with $k + \mathcal{O}(1)$ layers.

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