Paper Detail
YongShun Wang, JianLin Su, Yong Ma
Orthogonality in a LoRA factor does not by itself specify what the composed update protects: the answer depends on the task-start state, the parameterization, and the realized optimizer displacement. We formalize this question through Bilinear Optimization Divergence (BOD), an anchor-relative diagnostic of effective-update response on selected historical features. The finite-step analysis distinguishes two cases. In a shared adapter, protecting the routing displacement leaves a learned-anchor residual through the changing companion factor. In a fresh zero-output block, a feasible routing state can protect the composed update while both current factors remain trainable. These conditions yield Semi-Frozen Orthogonal Routing (SFOR) for shared adapters and current-block hard protection for cumulative O-LoRA; Weight Residual Projection (WRP) enforces the required displacement after the optimizer step. Controlled two-task traces verify the predicted residual paths, reducing normalized historical response from 19.12% to 0.005% in the shared family and from 7.72% to 0.002% in the cumulative family. Four-task experiments on Qwen3-8B characterize the resulting trade-offs: SFOR improves backward transfer (BWT) from -2.47 to -0.86 with nearly unchanged average accuracy (AA), while O-LoRA hard protection improves three-order mean AA from 80.27% to 81.30% and forgetting measure (FM) from 2.20 to 0.43. Component controls also show that stricter feasibility need not improve final task performance. Together, the analysis and evidence provide an architecture-conditioned account of which constraint to enforce, how to enforce it, and how to interpret its empirical value.
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@article{wang2026bilinear,
title = {Bilinear Optimization Divergence: Diagnosing Factor-Constrained LoRA Continual Learning},
author = {YongShun Wang and JianLin Su and Yong Ma},
year = {2026},
abstract = {Orthogonality in a LoRA factor does not by itself specify what the composed update protects: the answer depends on the task-start state, the parameterization, and the realized optimizer displacement. We formalize this question through Bilinear Optimization Divergence (BOD), an anchor-relative diagnostic of effective-update response on selected historical features. The finite-step analysis distinguishes two cases. In a shared adapter, protecting the routing displacement leaves a learned-anchor re},
url = {https://arxiv.org/abs/2609.23594},
keywords = {cs.LG},
eprint = {2609.23594},
archiveprefix = {arXiv},
}
{}