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arXiv · 2605.07463

Approximation Error Upper and Lower Bounds for H\"{o}lder Class with Transformers

Xin He·Yuling Jiao·Xiliang Lu·Jerry Zhijian Yang
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Abstract

We explore the expressive power of Transformers by establishing precise approximation error upper and lower bounds for H\"{o}lder class. Specifically, a new approximation upper bound is derived for the standard Transformer architecture equipped with Softmax operators, ReLU activation functions, and residual connections. We prove that a Transformer network composed of at most $\mathcal{O}(\varepsilon^{-{d_{0}}/{\alpha}})$ blocks can approximate any bounded H\"{o}lder function with $d_{0}$-dimensional input and smoothness $\alpha\in(0,1]$ under any accuracy $\varepsilon>0$. In the case of approximation lower bounds, leveraging the VC-dimension upper bound, we are the first to rigorously prove that Transformers demand for at least $\Omega(\varepsilon^{-{d_{0}}/({4\alpha})})$ blocks to achieve the $\varepsilon$ approximation accuracy. As a final step, we extend the derived results for standard Transformers to a general regression task and establish the corresponding excess risk rates demonstrating Transformers' empirical effectiveness in real-world settings.

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Xin He, Yuling Jiao, Xiliang Lu, Jerry Zhijian Yang. 2026-05-08. Approximation Error Upper and Lower Bounds for H\"{o}lder Class with Transformers. https://arxiv.org/abs/2605.07463

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