Catastrophic Forgetting is Low-Rank: A Function-Space Theory for Continual Adaptation

Ido Nitzan Hidekel; Dan Raviv

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arXiv · 2606.18024

Catastrophic Forgetting is Low-Rank: A Function-Space Theory for Continual Adaptation

Ido Nitzan Hidekel·Dan Raviv

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Abstract

Catastrophic forgetting in continual adaptation is usually studied through parameter drift, replay, or distillation, but these views do not identify which output-space directions are vulnerable. We give a function-space account in the NTK regime: new-task training induces old-task prediction drift through the cross-task kernel, yielding a closed-form predictor for the forgetting vector before any new-task gradient step. In frozen-backbone linear-head PEFT-CL, where the model is linear in the trainable parameters, the predictor is exact up to numerical precision; for nonlinear adapters/full fine-tuning, it is a local NTK approximation. The same expression reveals that forgetting concentrates in a small number of old-task NTK eigenmodes and under frozen linear heads gives a Kronecker scaling rule for the vulnerable rank. These results clarify the relation to prior NTK-overlap theory, explain why parameter-space regularizers can miss output-space interference, and motivate a targeted spectral regularizer.

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BibTeX RIS

Ido Nitzan Hidekel, Dan Raviv. 2026-06-16. Catastrophic Forgetting is Low-Rank: A Function-Space Theory for Continual Adaptation. https://arxiv.org/abs/2606.18024

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