Collective Kernel EFT for Pre-activation ResNets

Abstract

In finite-width deep neural networks, the empirical kernel $G$ evolves stochastically across layers. We develop a collective kernel effective field theory (EFT) for a simplified pre-activation residual model. Exact conditional Gaussianity of the residual increment gives finite-width discrete identities for $G$ and its sigma-kernel source. A projected $G$-only closure and a continuous-depth approximation define reference equations for the mean kernel $K_0$, the kernel covariance $V_4$, and the next-to-leading mean correction $K_1$. Numerical tests show that this closure captures the leading mean-kernel evolution, whereas its covariance and next-to-leading accuracy depend on sigma-kernel modes omitted from the $G$-only state. For the scalar ($N=1$) model with $\tanh$ activation, a Hermite--Galerkin extension agrees with the extrapolated $K_1$ value and reduces the $V_4$ endpoint error by $85$--$93\%$ relative to the $G$-only reference. These results clarify the range of validity of the equal-time Markov $G$-only description and identify the additional collective information required beyond it.

arXiv comment45 pages. v2: major improvements. Published version