Inter-Layer Hessian Analysis of Neural Networks with DAG Architectures

TL;DR

This paper introduces a formalism to decompose the Hessian of neural networks into inter-layer blocks, revealing structural curvature interactions and providing diagnostic metrics for understanding network geometry.

Contribution

It presents an analytical decomposition of the Hessian for DAG-structured networks, along with new metrics to analyze layer interactions and residual curvature effects.

Findings

Hessian decomposition separates convex and residual components.

Tensor component of input Hessian vanishes for ReLU activations.

Empirical validation on MLPs and ResNet-18 confirms theoretical insights.

Abstract

Modern automatic differentiation frameworks (JAX, PyTorch) return the Hessian of the loss function as a monolithic tensor, without exposing the internal structure of inter-layer interactions. This paper presents an analytical formalism that explicitly decomposes the full Hessian into blocks indexed by the DAG of an arbitrary architecture. The canonical decomposition $H = H^{GN} + H^T$ separates the Gauss--Newton component (convex part) from the tensor component (residual curvature responsible for saddle points). For piecewise-linear activations (ReLU), the tensor component of the input Hessian vanishes ($H^{T}_{v,w}\!\equiv\!0$ a.e., $H^f_{v,w}\!=\!H^{GN}_{v,w}\!\succeq\!0$); the full parametric Hessian contains residual terms that do not reduce to the GGN. Building on this decomposition, we introduce diagnostic metrics (inter-layer resonance~$\mathcal{R}$, geometric coupling~$\mathcal{C}$, stable rank~$\mathcal{D}$, GN-Gap) that are estimated stochastically in $O(P)$ time and reveal structural curvature interactions between layers. The theoretical analysis explains exponential decay of resonance in vanilla networks and its preservation under skip connections; empirical validation spans fully connected MLPs (Exp.\,1--5) and convolutional architectures (ResNet-18, ${\sim}11$M~parameters, Exp.\,6). When the architecture reduces to a single node, all definitions collapse to the standard Hessian $\nabla^2_\theta\mathcal{L}(\theta)\in\mathbb{R}^{p\times p}$.