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

2026-09-08

AI Summary

该研究通过在函数空间中直接开发统计力学描述来探讨深度神经网络的学习动态问题，使用了学习算子和统计算子等方法来解析误差动态和参数空间的状态密度。

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 $$ Φ_{\mathrm{fluc}}(M;B)=\frac{σ_ξ^2}{2}\log\det(M^{-1}+B)+\mathrm{const}. $$ 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, \(B=σ_ξ^2L^\ast\mathcal K L\), 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.

Problem

• function space
• statistical mechanics
• learning dynamics
• deep neural networks

Innovation

• function space
• statistical mechanics
• learning dynamics
• macroscopic variables
• dynamical Boltzmann weight