Making LoRA Identifiable: Orthogonal Alignment for Continual Learning

There is growing interest in applying Parameter-Efficient Fine-Tuning (PEFT) to Continual Learning, as it enables faster training, mitigates catastrophic forgetting, and facilitates better adaptation to new tasks. Prior approaches have explored a range of strategies, including extending new LoRA branches for each incoming task, training under orthogonal loss objectives across all task-specific LoRA modules, and employing gating mechanisms to integrate new and existing LoRA modules. However, these methods incur memory costs that grow linearly with the number of tasks, limiting their scalability. More recent approaches investigate training a single LoRA by leveraging asymmetries in its parameterization; however, they often suffer from insufficient representational alignment across the LoRA parameter space. To address these limitations, we propose Stiefel Optimization and Aligned Rotation (SOAR), a two-stage framework for continual LoRA learning. First, we constrain the $\mathbf{A}$ matrix to lie on the Stiefel manifold, thereby reducing coordinate misalignment to the space of orthogonal rotation matrices. Second, we introduce an Orthogonal Procrustes alignment procedure for the $\mathbf{B}$ matrix to estimate the optimal rotation, coupled with a per-rank retention merging strategy that quantifies directional agreement between representations of new and previously learned tasks. We conduct extensive experiments across multiple models and benchmarks, demonstrating that SOAR achieves strong continual learning performance without incurring additional memory overhead.