SAD-LoRA: Spectral Alignment for Low-Rank Knowledge Distillation

arXiv:2607.04306v1 Announce Type: cross Abstract: Distilling a fine-tuned teacher into a LoRA-adapted student is a standard recipe for parameter-efficient compression, but output-level KD does not explicitly control which rank-r weight subspace the adapter occupies. We propose extbf{SAD-LoRA} (extbf{S}pectral extbf{A}lignment extbf{D}istillation), which selects this subspace from the data-weighted student-space reference update DWTSigx^{1/2} and maintains it during training via a differentiable principal-angle loss on olspan(B). We show that the data-weighted distillation error decomposes exactly into subspace misalignment, within-subspace coefficient mismatch, and irreducible rank residual; standard KD can affect the first term only indirectly through output gradients. On controlled synthetic problems with a flat teacher spectrum, SAD-LoRA reduces the subspace-misalignment term from 51% to nearly zero and lifts final subspace alignment from 0.49 to 1.00. On RoBERTa-large to RoBERTa-base distillation across six GLUE tasks, SAD-LoRA improves rank efficiency: at r{=}4, it matches or beats the strongest included spectral baseline on five of six tasks, and at r{=}8 it gives the best result on SST-2 and CoLA. Ablations identify subspace alignment as the load-bearing component, while coefficient matching is auxiliary.

Source: arXiv cs.CV | 2026-07-07