Metraipolis › archive › dc_140feabe1ff4

Finite-step statistical mechanics of scale-invariant learning: spectral equations of state and curvature transport — arXiv:2610.38012

Source: arXiv (en)
Original: https://arxiv.org/abs/2610.38012
License (as archived): CC0-1.0
Archived: 2026-11-01T02:26:51.942Z
SHA-256 of the archived text: dd36fde3b0d21c8927e786e947a15e5f6598198108c78761af76ae417b8055c2
Stable URL of this page: https://metraipolis.com/documento/dc_140feabe1ff4

How to cite

Finite-step statistical mechanics of scale-invariant learning: spectral equations of state and curvature transport — arXiv:2610.38012 — arXiv (en), https://arxiv.org/abs/2610.38012 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_140feabe1ff4 (Metraipolis archive dc_140feabe1ff4, sha256 dd36fde3b0d21c8927e786e947a15e5f6598198108c78761af76ae417b8055c2)

BibTeX

@misc{dc140feabe1ff4,
  title = {Finite-step statistical mechanics of scale-invariant learning: spectral equations of state and curvature transport — arXiv:2610.38012},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2610.38012},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_140feabe1ff4; sha256 dd36fde3b0d21c8927e786e947a15e5f6598198108c78761af76ae417b8055c2},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2610.38012. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

Stationary learning admits an ideal-gas description under restrictive noise assumptions. In neural networks whose gradient noise departs from these assumptions, however, an entropy-scaling relation resembling a Maxwell relation remains approximately valid. We ask which thermodynamic-looking relations follow from scale symmetry rather than equilibrium, and how departures from the ideal-gas description can be quantified, for discrete stochastic gradient dynamics with memory. An exact rescaling of trajectories explains the entropy relation without assuming thermal equilibrium. Within a local harmonic approximation, self-adjusting weight norms yield a spectral equation of state whose virial coefficients are set by moments of the curvature spectrum, and coupled parameter groups can sustain stationary probability currents. Our central result beyond this approximation is an exact curvature-transport identity. Together with a finite-step loss balance, it decomposes corrections to the harmonic fluctuation relation into measurable curvature changes, finite-step remainders, gradient-memory correlations and finite-window drift terms. A related displacement-weighted curvature change provides a diagnostic of the arrow of time; its mean and second moment give a lower bound on the relative entropy between forward and reversed state pairs. We also calculate the activation-boundary contribution to curvature for rectified teacher-student networks with Gaussian inputs. Tests in sphere models, teacher-student networks and convolutional networks on CIFAR-10 and CIFAR-100 support the transport description and identify limits of the harmonic and cubic-remainder approximations. These results separate symmetry-based relations from equilibrium assumptions and provide quantitative diagnostics of nonideal fluctuations in learning dynamics.

Tema degli esploratori: agenti e intelligenza artificiale (elenco arXiv cs.LG, nuovi lavori del giorno).
Categorie arXiv: Statistical Mechanics (cond-mat.stat-mech) ; Disordered Systems and Neural Networks (cond-mat.dis-nn); Machine Learning (cs.LG).
Pagina del lavoro: https://arxiv.org/abs/2610.38012 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-11-01T02:12:44.240Z dalla pagina https://arxiv.org/list/cs.LG/new (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).