Efficient and Minimax-optimal In-context Nonparametric Regression with Transformers

• URL: http://arxiv.org/abs/2601.15014v1
• Date: Wed, 21 Jan 2026 14:13:38 GMT
• Title: Efficient and Minimax-optimal In-context Nonparametric Regression with Transformers
• Authors: Michelle Ching, Ioana Popescu, Nico Smith, Tianyi Ma, William G. Underwood, Richard J. Samworth,
• Abstract summary: We prove that a pretrained transformer with $(log n) parameters and $bigl(n2/(2+d)log3 nbigr)$ pretraining sequences can achieve the minimax-optimal rate of convergence.
• Score: 5.687100661457289
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We study in-context learning for nonparametric regression with $α$-Hölder smooth regression functions, for some $α>0$. We prove that, with $n$ in-context examples and $d$-dimensional regression covariates, a pretrained transformer with $Θ(\log n)$ parameters and $Ω\bigl(n^{2α/(2α+d)}\log^3 n\bigr)$ pretraining sequences can achieve the minimax-optimal rate of convergence $O\bigl(n^{-2α/(2α+d)}\bigr)$ in mean squared error. Our result requires substantially fewer transformer parameters and pretraining sequences than previous results in the literature. This is achieved by showing that transformers are able to approximate local polynomial estimators efficiently by implementing a kernel-weighted polynomial basis and then running gradient descent.

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