Leveraging tails for adaptation

2026-06-18 · 16:57 UTC ·Sergios Agapiou, Ismaël Castillo, Paul Egels ·1 min read

Brief

Agapiou, Castillo, and Egels analyze Bayesian posterior contraction in nonparametric models when coefficients receive p-exponential priors (Laplace p=1; p<1 heavier tails). They prove contraction rates strictly improve as p decreases and establish near-full adaptation to unknown smoothness (up to log factors) in a p→0 regime. Applications include series priors in white-noise regression and overparametrized shallow ReLU networks adapting to any regularity 0≤β≤2; simulations corroborate theory.

Abstract

We consider contraction of Bayesian posterior distributions in nonparametric settings where coefficients of a function over a basis or dictionary are given priors with $p$--exponential tails, including Laplace tails $(p=1)$ and heavier tails $(p<1)$. It is shown that contraction rates improve as $p$ decreases and that full adaptation to smoothness, up to logarithmic factors, is obtained in an appropriate $p\to 0$ regime. As applications, we consider both series priors in white noise regression and shallow ReLU neural networks in random design regression. In particular, we show that overparametrised shallow ReLU networks can adapt to any regularity $0\le β\le 2$. Through a simulation study, we show strong empirical agreement with the behavior predicted by our theory.

Comment: 59 pages, 3 figures