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 $P_X$ on $[0,1]^d$, we obtain contraction in Rényi divergence and, after clipping posterior draws at a bound on the target range, in squared $\mathcal{L}^2(P_X)$ 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.