arXiv subjects

Maxim Bolshim

Publications and source records attributed to Maxim Bolshim.

4 recordsLinked to original sources

arXiv2026

Profile-aware diagnostics of discarded parameter-Hessian mass

A discarded parameter-Hessian fraction depends on the diagonal profile and parameter partition; it is not the Fisher information discarded by K-FAC. We compare off-block squared Frobenius mass with group-size and measured diagonal-profile references at specified weights, inputs and coordinates. Similar initial fractions in ResNet-18 and its skip-free twin accompany different profiles. On the same saved trained ResNet-18 states, changing from stage groups to layers moves estimated mass from below to above the profile envelope. An approximate joint conditional Monte Carlo screen resolves this crossing in 6/6 seed/input cells; the skip-free network has 2 resolved and 4 unresolved comparisons. A separate comparison of nine saved states under two geometry draws splits the full model residual into off-group Hessian action and its remainder: gradient, within-group action and damping. At unit correction scale, off-group input alone increases loss in 17 of 18 state-geometry comparisons; in nine of those cases, the full candidate has lower loss than the remainder-only candidate. After correction-norm matching, seven of the ten comparisons that initially favoured full favour the remainder. These finite-loss contrasts depend on scale and do not establish a restoration benefit.

cs.LG↗

arXiv2026

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 $16/120$.

cs.LG↗

arXiv2026

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

cs.LG↗

arXiv2025

Local properties of neural networks through the lens of layer-wise Hessians

We introduce a methodology for analyzing neural networks through the lens of layer-wise Hessian matrices. The local Hessian of each functional block (layer) is defined as the matrix of second derivatives of a scalar function with respect to the parameters of that layer. This concept provides a formal tool for characterizing the local geometry of the parameter space. We show that the spectral properties of local Hessians, such as the distribution of eigenvalues, reveal quantitative patterns associated with overfitting, underparameterization, and expressivity in neural network architectures. We conduct an extensive empirical study involving 111 experiments across 37 datasets. The results demonstrate consistent structural regularities in the evolution of local Hessians during training and highlight correlations between their spectra and generalization performance. These findings establish a foundation for using local geometric analysis to guide the diagnosis and design of deep neural networks. The proposed framework connects optimization geometry with functional behavior and offers practical insight for improving network architectures and training stability.

cs.LG↗