In-Context Learning of Linear Dynamical Systems with Transformers: Error Bounds and Depth-Separation

• URL: http://arxiv.org/abs/2502.08136v2
• Date: Fri, 14 Feb 2025 04:54:41 GMT
• Title: In-Context Learning of Linear Dynamical Systems with Transformers: Error Bounds and Depth-Separation
• Authors: Frank Cole, Yulong Lu, Tianhao Zhang, Yuxuan Zhao,
• Abstract summary: This paper investigates approximation-theoretic aspects of the in-context learning capability of the transformers in representing a family of noisy linear dynamical systems. Our first theoretical result establishes an upper bound on the approximation error of multi-layer transformers with respect to an $L2$-testing loss uniformly defined across tasks. Our second result establishes a non-diminishing lower bound on the approximation error for a class of single-layer linear transformers.
• Score: 16.748746646611412
• License:
• Abstract: This paper investigates approximation-theoretic aspects of the in-context learning capability of the transformers in representing a family of noisy linear dynamical systems. Our first theoretical result establishes an upper bound on the approximation error of multi-layer transformers with respect to an $L^2$-testing loss uniformly defined across tasks. This result demonstrates that transformers with logarithmic depth can achieve error bounds comparable with those of the least-squares estimator. In contrast, our second result establishes a non-diminishing lower bound on the approximation error for a class of single-layer linear transformers, which suggests a depth-separation phenomenon for transformers in the in-context learning of dynamical systems. Moreover, this second result uncovers a critical distinction in the approximation power of single-layer linear transformers when learning from IID versus non-IID data.

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