提出自适应选择共享因子的联邦LoRA方法，提升模型微调效果

Low-rank adaptation (LoRA) represents large language model (LLM) updates with two compact matrix factors, i.e., $A$ and $B$, providing an efficient way to fine-tune large models in federated learning paradigm. Inspired by the asymmetric roles of the LoRA factors, we study whether $A$ should be shared across clients while $B$ remains client-specific (Share-A/Local-B), or whether $B$ should instead be shared while $A$ remains client-specific (Share-B/Local-A). With a least-squares surrogate, we reveal that Share-A/Local-B requires the client-specific LoRA update matrices to use a common rank-$r$ input-side space, whereas Share-B/Local-A requires a common rank-$r$ output-side space. The two strategies therefore incur different projection residuals, indicating that the preferred strategy is the one with the smaller aggregate residual across clients. With this insight, we propose Federated Adaptive Factor Sharing Low-Rank Adaptation (FedAS-LoRA), which selects the sharing side before training to enhance fine-tuning performance. To enable adaptive factor selection before training, we design a Rank-Aware Shared-Subspace Sufficiency (RSS) metric, which effectively assesses whether a shared rank-$r$ input subspace is sufficient for the local data distributions using representations extracted from a frozen LLM backbone. Experiments across different tasks, data distributions, LoRA ranks, and participation settings confirm the effectiveness of RSS and the superior performance of FedAS-LoRA.