Renormalizable Spectral-Shell Dynamics as the Origin of Neural Scaling Laws

TL;DR

This paper derives a macroscopic spectral-shell framework from gradient descent in neural networks, explaining neural scaling laws and double descent phenomena through a unified, self-similar dynamical model.

Contribution

It introduces a renormalizable spectral-shell dynamics approach that unifies lazy training and feature learning, providing a theoretical foundation for neural scaling laws.

Findings

Explains neural scaling laws and double descent phenomena.

Unifies lazy training and feature learning.

Provides explicit scaling exponents and self-similar solutions.

Abstract

Neural scaling laws and double-descent phenomena suggest that deep-network training obeys a simple macroscopic structure despite highly nonlinear optimization dynamics. We derive such structure directly from gradient descent in function space. For mean-squared error loss, the training error evolves as $\dot e_t=-M(t)e_t$ with $M(t)=J_{\theta(t)}J_{\theta(t)}^{\!*}$, a time-dependent self-adjoint operator induced by the network Jacobian. Using Kato perturbation theory, we obtain an exact system of coupled modewise ODEs in the instantaneous eigenbasis of $M(t)$. To extract macroscopic behavior, we introduce a logarithmic spectral-shell coarse-graining and track quadratic error energy across shells. Microscopic interactions within each shell cancel identically at the energy level, so shell energies evolve only through dissipation and external inter-shell interactions. We formalize this via a \emph{renormalizable shell-dynamics} assumption, under which cumulative microscopic effects reduce to a controlled net flux across shell boundaries. Assuming an effective power-law spectral transport in a relevant resolution range, the shell dynamics admits a self-similar solution with a moving resolution frontier and explicit scaling exponents. This framework explains neural scaling laws and double descent, and unifies lazy (NTK-like) training and feature learning as two limits of the same spectral-shell dynamics.