PreprintCC BY 4.0arXiv
Sieve-free ReLU Bayesian priors with near-minimax posterior contraction
Abstract
We establish fractional-posterior contraction for shallow ReLU networks in Gaussian random-design regression with a β-Hölder target, using priors independent of the sample size and smoothness parameters. For any fixed design law PX on [0,1]d, we obtain contraction in Rényi divergence and, after clipping posterior draws at a bound on the target range, in squared ℒ2(PX) loss at rate (log n/n)2β/(2β+d). This matches the classical Hölder minimax rate up to a logarithmic factor. These priors have random finite width and use width-dependent uniform or Gaussian weight distributions, or width-independent Student t distributions, including Cauchy distributions.
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BibTeX
@misc{nguyen2026sievefree,
title = {{Sieve-free ReLU Bayesian priors with near-minimax posterior contraction}},
author = {Hien Duy Nguyen and Jacob Westerhout and Julyan Arbel},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.23308},
eprint = {2610.23308},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.23308},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sieve-free ReLU Bayesian priors with near-minimax posterior contraction
AU - Hien Duy Nguyen
AU - Jacob Westerhout
AU - Julyan Arbel
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.23308
UR - https://arxiv.org/abs/2610.23308
AB - We establish fractional-posterior contraction for shallow ReLU networks in Gaussian random-design regression with a β-Hölder target, using priors independent of the sample size and smoothness parameters. For any fixed design law P_X on [0,1]^d, we obtain contraction in Rényi divergence and, after clipping posterior draws at a bound on the target range, in squared ℒ^2(P_X) loss at rate (log n/n)^2β/(2β+d). This matches the classical Hölder minimax rate up to a logarithmic factor. These priors have random finite width and use width-dependent uniform or Gaussian weight distributions, or width-independent Student t distributions, including Cauchy distributions.
ER -