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arXiv · 2504.15956

Universal Approximation with Softmax Attention

Jerry Yao-Chieh Hu·Hude Liu·Hong-Yu Chen·Weimin Wu·Han Liu
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Abstract

We prove that with linear transformations, both (i) two-layer self-attention and (ii) one-layer self-attention followed by a softmax function are universal approximators for continuous sequence-to-sequence functions on compact domains. Our main technique is a new interpolation-based method for analyzing attention's internal mechanism. This leads to our key insight: self-attention is able to approximate a generalized version of ReLU to arbitrary precision, and hence subsumes many known universal approximators. Building on these, we show that two-layer multi-head attention alone suffices as a sequence-to-sequence universal approximator. In contrast, prior works rely on feed-forward networks to establish universal approximation in Transformers. Furthermore, we extend our techniques to show that, (softmax-)attention-only layers are capable of approximating various statistical models in-context. We believe these techniques hold independent interest.

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Jerry Yao-Chieh Hu, Hude Liu, Hong-Yu Chen, Weimin Wu, Han Liu. 2025-04-22. Universal Approximation with Softmax Attention. https://arxiv.org/abs/2504.15956

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