BaryLoRA: State-Conditioned Product Barycenters for Federated Low-Rank Adaptation

Federated LoRA provides a communication-efficient way to adapt pretrained language models, but a shared adapter still requires the server to decide both what to aggregate and how to assign client contributions. Averaging low-rank factors can fail to aggregate the effective updates they encode: equivalent factorizations are non-unique, and averaging the factors separately introduces cross-client products. Meanwhile, sample-size weighting assigns each client the same relative mass at every adapted layer and therefore does not directly specify client-layer masses for a shared effective-update consensus. We introduce BaryLoRA, a state-conditioned product-space consensus framework that addresses these two mismatches jointly. BaryLoRA treats normalized effective updates as the consensus object and uses persistent optimization state to tilt a sample-size prior into client-layer contribution masses. The server then forms a weighted mean of the effective updates and returns its best rank-constrained approximation through a factorized computation that preserves the communication efficiency of low-rank adaptation. We distinguish the empirical state-to-mass design from the conditional properties of the resulting consensus. Under fixed positive state transforms and contribution masses, we establish uniqueness of the mass solution, invariance to equivalent LoRA factorizations, and optimality of the factorized rank-constrained realization. Experiments on language understanding, code generation, and mathematical reasoning tasks evaluate BaryLoRA against federated LoRA baselines.