A Function-Space Approach to the Statistical Mechanics of Learning Dynamics
Updated 5 h ago · first seen 11 Sept 2026
paper_01M294GK5DHZNPR8G2CR892Z9R
- Published
- 11 Sept 2026
- T1 · 5 h ago
- arXiv
- 2609.09589
- T1 · 5 h ago
- Category
- cs.AI
- T1 · 5 h ago
Abstract
Deep neural networks exhibit regular macroscopic behavior despite highly nonlinear dynamics in vast parameter spaces. We develop a statistical-mechanical description of learning directly in function space, treating parameter configurations as microscopic realizations and functions with their dynamical operators as macroscopic variables. For mean-squared loss, the exact error dynamics are governed by the learning operator \(M=JJ^\ast\). Combining the dynamical Boltzmann weight of the conditional stochastic dynamics with the parameter-space density of states, whose local curvature defines a statistical operator \(B\), and integrating over local fluctuations yields $$ \Phi_{\mathrm{fluc}}(M;B)=\frac{\sigma_\xi^2}{2}\log\det(M^{-1}+B)+\mathrm{const}. $$ At fixed spectrum, this term is rotationally stationary when \([M,B]=0\), is minimized by pairing large eigenvalues of \(M\) with small eigenvalues of \(B\), and generates a local restoring contribution against rotational mismatch. For ReLU-type function spaces under mild stable statistical conditions, \(B=\sigma_\xi^2L^\ast\mathcal K L\), where \(L\) measures coarse-grained second-order structure. Thus the low-\(B\) sector corresponds, up to bounded anisotropy of \(\mathcal K\), to low structural curvature, implying a preference for faster relaxation along smooth, data-adaptive directions. These results identify function space as a natural macroscopic level for studying stable collective organization in learning.
Authors 4
Yizhou Zhang, Weichen Wu, Lun Du, Zhengjie Miao
Specification
- Official page
Source:arXiv (Atom API + RSS)T1observed 5 h agohigh
- Arxiv announce type
- new
Source:arXiv (Atom API + RSS)T1observed 5 h agohigh
- arXiv id
- 2609.09589
Source:arXiv (Atom API + RSS)T1observed 5 h agohigh
- Categories
- cs.AI
Source:arXiv (Atom API + RSS)T1observed 5 h agohigh
Source:arXiv (Atom API + RSS)T1observed 5 h agohigh
- Primary category
- cs.AI
Source:arXiv (Atom API + RSS)T1observed 5 h agohigh
- Published
- 11 Sept 2026
Source:arXiv (Atom API + RSS)T1observed 5 h agohigh
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Attributed facts
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T19
Freshest observation
5 h ago
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- Authors
- Yizhou Zhang, Weichen Wu, Lun Du
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Claim history
Official pageofficial_url1
| Value | Valid from → to | Status | Source | Confidence | Extractor |
|---|---|---|---|---|---|
| https://arxiv.org/abs/2609.09589 | → current | current | arXiv (Atom API + RSS)T1 | high | deterministic |
Abstractabstract1
| Value | Valid from → to | Status | Source | Confidence | Extractor |
|---|---|---|---|---|---|
| Deep neural networks exhibit regular macroscopic behavior despite highly nonlinear dynamics in vast parameter spaces. We develop a statistical-mechanical description of learning directly in function space, treating parameter configurations as microscopic realizations and functions with their dynamical operators as macroscopic variables. For mean-squared loss, the exact error dynamics are governed by the learning operator \(M=JJ^\ast\). Combining the dynamical Boltzmann weight of the conditional stochastic dynamics with the parameter-space density of states, whose local curvature defines a statistical operator \(B\), and integrating over local fluctuations yields $$ \Phi_{\mathrm{fluc}}(M;B)=\frac{\sigma_\xi^2}{2}\log\det(M^{-1}+B)+\mathrm{const}. $$ At fixed spectrum, this term is rotationally stationary when \([M,B]=0\), is minimized by pairing large eigenvalues of \(M\) with small eigenvalues of \(B\), and generates a local restoring contribution against rotational mismatch. For ReLU-type function spaces under mild stable statistical conditions, \(B=\sigma_\xi^2L^\ast\mathcal K L\), where \(L\) measures coarse-grained second-order structure. Thus the low-\(B\) sector corresponds, up to bounded anisotropy of \(\mathcal K\), to low structural curvature, implying a preference for faster relaxation along smooth, data-adaptive directions. These results identify function space as a natural macroscopic level for studying stable collective organization in learning. | → current | current | arXiv (Atom API + RSS)T1 | high | deterministic |
Arxiv announce typearxiv_announce_type1
| Value | Valid from → to | Status | Source | Confidence | Extractor |
|---|---|---|---|---|---|
| new | → current | current | arXiv (Atom API + RSS)T1 | high | deterministic |
arXiv idarxiv_id1
| Value | Valid from → to | Status | Source | Confidence | Extractor |
|---|---|---|---|---|---|
| 2609.09589 | → current | current | arXiv (Atom API + RSS)T1 | high | deterministic |
Categoriescategories1
| Value | Valid from → to | Status | Source | Confidence | Extractor |
|---|---|---|---|---|---|
| cs.AI | → current | current | arXiv (Atom API + RSS)T1 | high | deterministic |
PDFpdf_url1
| Value | Valid from → to | Status | Source | Confidence | Extractor |
|---|---|---|---|---|---|
| https://arxiv.org/pdf/2609.09589 | → current | current | arXiv (Atom API + RSS)T1 | high | deterministic |
Primary categoryprimary_category1
| Value | Valid from → to | Status | Source | Confidence | Extractor |
|---|---|---|---|---|---|
| cs.AI | → current | current | arXiv (Atom API + RSS)T1 | high | deterministic |
Publishedpublished_at1
| Value | Valid from → to | Status | Source | Confidence | Extractor |
|---|---|---|---|---|---|
| 11 Sept 2026 | → current | current | arXiv (Atom API + RSS)T1 | high | deterministic |
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New paper: A Function-Space Approach to the Statistical Mechanics of Learning Dynamics
arxiv
| Source | Document | Type | Tier | Last observed | Snapshots |
|---|---|---|---|---|---|
| arXiv (Atom API + RSS) | rss.arxiv.org/rss/cs.AI | feed | T1· Official | 5 h ago | 1 |
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