One-Step Curvature Probes Miss the Fitting Operator: Retained Capacity and Terminal Null-Space Correction for Continual Learning

arXiv:2610.03952v1 Announce Type: new Abstract: 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 Delta_P=PJ^op(JPJ^op)^{-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 fitting floor and an algorithm-dependent null-space excess, motivating terminal null-space correction, which preserves linearized new-task outputs on its Jacobian batch. Controlled checks validate the local quadratic and show that projection can greatly reduce matched-norm curvature while barely changing terminal forgetting. Across common-threshold configurations, projection yields the larger signed old-task loss change in 73/99 matched pairs, retained fitting capacity falls with rank, and fixed-rank comparisons separate terminal forgetting even when probe gain is approximately matched. On Permuted MNIST and Split CIFAR-100, terminal correction decreases signed old-task loss in 166/180 method--dataset--seed pairs under the all-seed intention-to-correct analysis; because acceptance and outcome reporting use the same held-out split, this result is descriptive and test-conditioned. Overall, terminal cost depends jointly on retained fitting capacity, endpoint-direction curvature, and the null-space component selected by optimization.

Source: arXiv cs.LG | 2026-10-06

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