Stochastic Estimation of the Layer-wise Hessian Trace for Monitoring Neural-network Training

The loss and the norm of its gradient separate the healthy and the pathological regimes of neural-network training only weakly, whilst the curvature of the empirical risk differs qualitatively between them but is inaccessible explicitly at parameter counts $P\sim 10^{6}-10^{8}$. We present a stochastic estimator of the trace of the diagonal blocks of the Hessian matrix of the empirical risk of a neural network. The procedure combines the Hutchinson stochastic trace estimator with a single Hessian-vector product over the whole parameter vector and recovers unbiased estimates of every per-layer trace in one backward pass through the computational graph. We show that correctness under weight sharing requires the layer-wise Hessian to be assembled before the second differentiation: unrolling shared weights into independent coordinates introduces a systematic bias whose sign and magnitude are governed by the cross-instance blocks of the unrolled Hessian. A closed-form expression for the variance of the estimator at a fixed Hessian is derived, together with a decomposition of the total variance under the mini-batch sampling distribution. This decomposition yields a critical probe count $K^{\star}$ that balances the two sources of randomness and supports the practical recommendation $K\in[5,10]$ in the on-line monitoring regime. The estimator is applied to the detection of the label-memorisation regime of ResNet-18, ResNet-34, and VGG-11 on CIFAR-10 and CIFAR-100, where a calibrated cumulative-sum decision rule attains an empirical detection power of $179/180$ at a false-alarm rate of $16/120$.

Inter-Layer Hessian Analysis of Neural Networks with DAG Architectures

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 $\nabla^2_θ\mathcal{L}(θ)\in\mathbb{R}^{p\times p}$.