Critical initialization destabilizes higher input derivatives in wide scalar-input networks — Paper

Abstract

The edge-of-chaos condition preserves first-order input perturbations in wide randomly initialized networks, but physics-informed losses, score matching and derivative regularization depend on higher input derivatives. For smooth scalar-input fully connected networks, using a joint Gaussianity of the finite derivative jet that holds in the infinite-width limit at each fixed depth, we derive mean-field recursions through third order that are exact at the variance fixed point, with finite-depth corrections that decay geometrically. At criticality, the first-derivative variance is depth-invariant, whereas the second-derivative variance grows linearly whenever the activation has nonzero curvature. The resulting third-order system closes on mean-field susceptibilities. For residual networks with branch scale L^{-1/2}, we prove that every fixed finite derivative order has uniformly bounded variance under explicit regularity assumptions. Simulations verify the critical growth laws, the residual bound, and the closed recursion. The results concern initialization, not trained-network performance.

Authors 2

Prashant Singh, Pranav Singh

Specification

Official page: arxiv.org/abs/2609.09244
Source:arXiv (Atom API + RSS)T1observed 5 h agohigh
Arxiv announce type: cross
Source:arXiv (Atom API + RSS)T1observed 5 h agohigh
arXiv id: 2609.09244
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Categories: stat.ML, cs.AI, cs.LG
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PDF: arxiv.org/pdf/2609.09244
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Primary category: stat.ML
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Published: 11 Sept 2026
Source:arXiv (Atom API + RSS)T1observed 5 h agohigh

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5 h ago

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Authors: Prashant Singh, Pranav Singh

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Official pageofficial_url1

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https://arxiv.org/abs/2609.09244	11 Sept 2026 21:03 UTC → current	current	arXiv (Atom API + RSS)T1	high	deterministic

Abstractabstract1

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The edge-of-chaos condition preserves first-order input perturbations in wide randomly initialized networks, but physics-informed losses, score matching and derivative regularization depend on higher input derivatives. For smooth scalar-input fully connected networks, using a joint Gaussianity of the finite derivative jet that holds in the infinite-width limit at each fixed depth, we derive mean-field recursions through third order that are exact at the variance fixed point, with finite-depth corrections that decay geometrically. At criticality, the first-derivative variance is depth-invariant, whereas the second-derivative variance grows linearly whenever the activation has nonzero curvature. The resulting third-order system closes on mean-field susceptibilities. For residual networks with branch scale L^{-1/2}, we prove that every fixed finite derivative order has uniformly bounded variance under explicit regularity assumptions. Simulations verify the critical growth laws, the residual bound, and the closed recursion. The results concern initialization, not trained-network performance.	11 Sept 2026 21:03 UTC → current	current	arXiv (Atom API + RSS)T1	high	deterministic

Arxiv announce typearxiv_announce_type1

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cross	11 Sept 2026 21:03 UTC → current	current	arXiv (Atom API + RSS)T1	high	deterministic

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2609.09244	11 Sept 2026 21:03 UTC → current	current	arXiv (Atom API + RSS)T1	high	deterministic

Authorsauthors1

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Prashant Singh, Pranav Singh	11 Sept 2026 21:03 UTC → current	current	arXiv (Atom API + RSS)T1	high	deterministic

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stat.ML, cs.AI, cs.LG	11 Sept 2026 21:03 UTC → current	current	arXiv (Atom API + RSS)T1	high	deterministic

PDFpdf_url1

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https://arxiv.org/pdf/2609.09244	11 Sept 2026 21:03 UTC → current	current	arXiv (Atom API + RSS)T1	high	deterministic

Primary categoryprimary_category1

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stat.ML	11 Sept 2026 21:03 UTC → current	current	arXiv (Atom API + RSS)T1	high	deterministic

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11 Sept 2026	11 Sept 2026 21:03 UTC → current	current	arXiv (Atom API + RSS)T1	high	deterministic

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11 Sept 2026
New paperPaperCritical initialization destabilizes higher input derivatives in wide scalar-input networks
New paper: Critical initialization destabilizes higher input derivatives in wide scalar-input networks
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arXiv (Atom API + RSS)	rss.arxiv.org/rss/cs.AI	feed	T1· Official	5 h ago	1

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