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 $16/120$.