Stable-Plastic Subspace Construction for Differentially Private Continual LoRA

Continual learning (CL) adapts pretrained models to a sequence of tasks while retaining knowledge acquired from earlier tasks. When task data are sensitive, differential privacy (DP) provides record-level protection, but maintaining both adaptation and retention under privacy constraints remains challenging. We study this problem in continual low-rank adaptation (LoRA) and observe that: 1) private training increases forgetting primarily during subsequent adaptation rather than initial task acquisition; 2) jointly perturbing both LoRA factors introduces a non-negligible bilinear noise interaction; and 3) DP aggravates the stability-plasticity dilemma, hindering both knowledge retention and adaptation to new tasks. Based on these observations, we propose DP-SPA, which constructs a stable subspace from previously privatized updates and a plastic subspace from pretrained weights within its orthogonal complement. The method fixes these bases and privately optimizes only the LoRA factor, eliminating the bilinear noise interaction within each step. Before merging each task update, it rescales stable-direction coefficients using historical and current update energies. We establish a privacy guarantee and show that recalibration does not increase historical overlap and tightens a conditional upper bound on old-task loss increase. Experiments across five datasets, eight settings, and multiple privacy budgets show that DP-SPA improves average accuracy and reduces forgetting compared with the evaluated baselines.