Near-optimal estimates for the $\ell^p$-Lipschitz constants of deep random ReLU neural networks — Paper

Abstract

-cross Abstract: This paper studies the $\ell^p$-Lipschitz constants of ReLU neural networks $\Phi: \mathbb{R}^d \to \mathbb{R}$ with random parameters for $p \in [1,\infty]$. The distribution of the weights follows a variant of the He initialization. In the case of zero-bias networks, we derive high probability upper and lower bounds for wide networks that differ at most by a factor that is logarithmic in the network's depth. Remarkably, the behavior of the $\ell^p$-Lipschitz constant varies significantly between the regimes $ p \in [1,2) $ and $ p \in [2,\infty] $. For $p \in [2,\infty]$, the $\ell^p$-Lipschitz constant behaves similarly to $\Vert g\Vert_{p'}$, where $g \in \mathbb{R}^d$ is a $d$-dimensional standard Gaussian vector and $1/p + 1/p' = 1$. In contrast, for $p \in [1,2)$, the $\ell^p$-Lipschitz constant aligns more closely to $\Vert g \Vert_{2}$. We extend our analysis to networks with possibly non-zero biases drawn from arbitrary symmetric distributions. In this case, we obtain high probability upper and lower bounds that differ at most by a factor that is logarithmic in the network's width and linear in its depth.

Authors 5

Sjoerd Dirksen, Patrick Finke, Paul Geuchen, Dominik St\"oger, Felix Voigtlaender

Specification

Official page: arxiv.org/abs/2506.19695
Source:arXiv (Atom API + RSS)T1observed 6 h agohigh
Arxiv announce type: replace
Source:arXiv (Atom API + RSS)T1observed 6 h agohigh
arXiv id: 2506.19695
Source:arXiv (Atom API + RSS)T1observed 6 h agohigh
Categories: stat.ML, cs.LG, math.PR
Source:arXiv (Atom API + RSS)T1observed 6 h agohigh
PDF: arxiv.org/pdf/2506.19695
Source:arXiv (Atom API + RSS)T1observed 6 h agohigh
Primary category: stat.ML
Source:arXiv (Atom API + RSS)T1observed 6 h agohigh
Published: 11 Sept 2026
Source:arXiv (Atom API + RSS)T1observed 6 h agohigh

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9

Source tiers

T19

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6 h ago

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Authors: Sjoerd Dirksen, Patrick Finke, Paul Geuchen

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Value	Valid from → to	Status	Source	Confidence	Extractor
2506.19695	11 Sept 2026 21:02 UTC → current	current	arXiv (Atom API + RSS)T1	high	deterministic

Claims are temporal and append-only: a new observation closes the previous claim (valid_to) instead of overwriting it. Conflicting claims from different sources are kept side by side and flagged — never averaged. Methodology →

11 Sept 2026
New paperPaperNear-optimal estimates for the $\ell^p$-Lipschitz constants of deep random ReLU neural networks
New paper: Near-optimal estimates for the $\ell^p$-Lipschitz constants of deep random ReLU neural networks
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Source	Document	Type	Tier	Last observed	Snapshots
arXiv (Atom API + RSS)	rss.arxiv.org/rss/cs.LG	feed	T1· Official	4 h ago	1

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