Collective Kernel EFT for Pre-activation ResNets

• URL: http://arxiv.org/abs/2604.15742v1
• Date: Fri, 17 Apr 2026 06:31:50 GMT
• Title: Collective Kernel EFT for Pre-activation ResNets
• Abstract summary: In finite-width deep neural networks, the empirical kernel $G$ evolvesally across layers.<n>We develop a collective kernel effective field theory (EFT) for pre-activation ResNets based on a $G$-only closure hierarchy and diagnose its finite closure window.
• Score: 0.20625936401496234
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 pre-activation ResNets based on a $G$-only closure hierarchy and diagnose its finite validity window. Exploiting the exact conditional Gaussianity of residual increments, we derive an exact stochastic recursion for $G$. Applying Gaussian approximations systematically yields a continuous-depth ODE system for the mean kernel $K_0$, the kernel covariance $V_4$, and the $1/n$ mean correction $K_{1,\mathrm{EFT}}$, which emerges diagrammatically as a one-loop tadpole correction. Numerically, $K_0$ remains accurate at all depths. However, the $V_4$ equation residual accumulates to an $O(1)$ error at finite time, primarily driven by approximation errors in the $G$-only transport term. Furthermore, $K_{1,\mathrm{EFT}}$ fails due to the breakdown of the source closure, which exhibits a systematic mismatch even at initialization. These findings highlight the limitations of $G$-only state-space reduction and suggest extending the state space to incorporate the sigma-kernel.

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