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This paper introduces a curvature-aligned criterion, $\Delta_2^{(D)}$, for probing local loss-landscape stabilization under sample growth in neural networks, focusing on the top-$D$ eigenspace of the empirical Hessian. They prove that $\Delta_2^{(D)}$ maintains the $O(k^{-2})$ mean-squared rate of full-space criteria while reducing curvature dependence to the subspace dimension $D$, and derive scalable estimators. Experiments on a decoder-only transformer demonstrate that this curvature-aligned probe accurately reproduces the full-space signal using a small fraction of the parameter space, with significant speedups using a closed-form estimator.
By probing only the most relevant directions in the loss landscape, you can achieve the same signal as full-space probing with orders-of-magnitude speedups.
Local loss-landscape stabilization under sample growth is typically measured either pointwise or through isotropic averaging in the full parameter space. Despite practical value, both choices probe directions that contribute little to the dominant local deformation of strongly anisotropic neural landscapes. We recast stabilization as an observational problem and introduce a unified family of criteria parameterized by an aggregation order and a probing distribution; within this family we propose a curvature-aligned criterion $\Delta_2^{(D)}$ that probes the loss increment field in the top-$D$ eigenspace of the empirical Hessian near a trained solution. Solely from a local quadratic model, we prove that $\Delta_2^{(D)}$ preserves the $O(k^{-2})$ mean-squared rate of the full-space criterion while replacing ambient-dimension curvature dependence with dependence on the subspace dimension $D$; a corollary gives a closed-form spectral expression and a proposition identifies the top-$D$ eigenspace as extremal within the eigenspace-aligned family. We also derive scalable estimators based on Hessian-vector products, subspace Monte Carlo, and a closed-form Gaussian-moment proxy. On a decoder-only transformer, a curvature-aligned probe occupying a tiny fraction of parameter space already reproduces the full-space mean-squared signal to within numerical noise throughout the validated local regime, and the closed-form estimator is orders of magnitude faster than direct Monte Carlo after subspace construction.