A Function-Space Approach to the Statistical Mechanics of Learning Dynamics

arXiv:2609.09589v1 Announce Type: new 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.

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.

Artificial Intelligence (cs.AI)

arXiv:2609.09589 [cs.AI]

(or arXiv:2609.09589v1 [cs.AI] for this version)

https://doi.org/10.48550/arXiv.2609.09589 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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A Function-Space Approach to the Statistical Mechanics of Learning Dynamics

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