One-Step Curvature Probes Miss the Fitting Operator: Retained Capacity and Terminal Null-Space Correction for Continual Learning
A one-step curvature probe evaluates an initial direction, whereas continual learners are judged after reaching comparable new-task fit. In an overparameterized linearization, projected gradient descent converges to $Δ_P=PJ^\top(JPJ^\top)^{-1}r$, and its squared-displacement inflation is exactly the reciprocal of the retained fitting capacity $c_P(r)$. More generally, the terminal old-task quadratic ratio factorizes as $1/[c_P(r)G_{\rm end}(P,r)]$, where $G_{\rm end}$ compares curvature along endpoint directions. In the rank-one case, $G_{\rm end}$ equals the one-step probe gain; for multiple outputs, the two gains can differ. The terminal quadratic also separates into a curvature-optimal...
| Authors | Abu Sa-Adat Mohamed Moon-Im Al Ahsan, Ibne Farabi Shihab, Md Najmus Swaqeeb |
|---|---|
| Primary category | cs.LG |
| Published | 2 October 2026 |
| Last updated at source | 2 October 2026 |
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