Beyond Pointwise Replay: Pathwise Knowledge Consolidation for Continual Learning

Continual learning (CL) aims to acquire new knowledge without forgetting what was learned before. Replay-based methods mitigate forgetting by revisiting a small memory buffer, but they supervise each replayed sample only at isolated targets, its label or its stored logits. Such pointwise supervision leaves the model underconstrained in function space and permits sharp, localized updates that cause catastrophic forgetting. To address this limitation, we propose Pathwise Knowledge Consolidation (PKC), which turns each replayed sample into a continuum of prediction tasks indexed by a corruption level: the label is geodesically corrupted toward the stored logits, and a lightweight training-only conditioner produces a prediction for each corrupted label. This connects the two previously isolated replay targets by a continuous trajectory of supervision, so the model preserves not only what to predict but also how its predictions evolve. A variational objective couples target fidelity along the path with path smoothness; its Euler-Lagrange characterization balances target fitting against path curvature, and a Hölder-type bound shows that the smoothness energy controls prediction drift across corruption levels. Extensive experiments demonstrate that PKC improves CL accuracy over strong replay methods while substantially reducing forgetting at no additional inference cost. PKC further outperforms recent prompt-based methods.