{"slug": "a-function-space-approach-to-the-statistical-mechanics-of-learning-dynamics", "title": "A Function-Space Approach to the Statistical Mechanics of Learning Dynamics", "summary": "A new arXiv paper (2609.09589v1) develops a statistical-mechanical description of deep neural network 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 authors show the exact error dynamics are governed by the learning operator M=JJ*, and combining the dynamical Boltzmann weight with the parameter-space density of states yields the fluctuation term Φ_fluc(M;B)=(σ_ξ²/2)log det(M⁻¹+B)+const. The work identifies function space as a natural macroscopic level for studying stable collective organization in learning, with the low-B sector corresponding to low structural curvature and a preference for faster relaxation along smooth, data-adaptive directions.", "body_md": "arXiv:2609.09589v1 Announce Type: new \nAbstract: 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\n  $$ \\Phi_{\\mathrm{fluc}}(M;B)=\\frac{\\sigma_\\xi^2}{2}\\log\\det(M^{-1}+B)+\\mathrm{const}. $$\n  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.", "url": "https://wpnews.pro/news/a-function-space-approach-to-the-statistical-mechanics-of-learning-dynamics", "canonical_source": "https://arxiv.org/abs/2609.09589", "published_at": "2026-09-11 04:00:00+00:00", "updated_at": "2026-09-11 04:27:52.165560+00:00", "lang": "en", "topics": ["machine-learning", "neural-networks", "ai-research"], "entities": ["arXiv", "M=JJ*", "B=σ_ξ²L*KL", "ReLU"], "alternates": {"html": "https://wpnews.pro/news/a-function-space-approach-to-the-statistical-mechanics-of-learning-dynamics", "markdown": "https://wpnews.pro/news/a-function-space-approach-to-the-statistical-mechanics-of-learning-dynamics.md", "text": "https://wpnews.pro/news/a-function-space-approach-to-the-statistical-mechanics-of-learning-dynamics.txt", "jsonld": "https://wpnews.pro/news/a-function-space-approach-to-the-statistical-mechanics-of-learning-dynamics.jsonld"}}