Stabilizing the Dynamic Low-Rank Training

Training neural networks directly in a low-rank parameterization is an appealing route to reducing memory, compute, and storage simultaneously during both training and inference. Dynamic low-rank training (DLRT), which confines weights to a rank-$r$ manifold via the Galerkin projection of the gradient flow, is particularly attractive because it identifies efficient subnetworks on the fly without specialized initialization or post-factorization. However, DLRT fails to find trainable networks under high compression. In this paper, we derive the gradient flow of the best rank-$r$ approximation and point out that the offset of DLRT comes from a curvature-coupling term which is large and thus non-negligible under aggressive compression. Guided by this analysis, we propose a stable dynamic low-rank training method, named SDLRT, which maintains a lightweight compensation buffer that reinjects the top neglected singular directions. Additionally, we introduce a negative feedback on the truncation tolerance to stabilize each layer's rank. Experimentally, SDLRT reliably finds trainable subnetworks where DLRT collapses and as a PEFT adapter on DeBERTa-v3, it achieves the best average score on SuperGLUE at only $2.8\\%$ parameter overhead over LoRA.