Representation-Correct Multi-Objective LoRA

Modern post-training often requires balancing multiple objectives, such as helpfulness and safety, while low-rank adaptation (LoRA) provides a parameter-efficient mechanism for model adaptation. However, LoRA factors are non-identifiable: infinitely many factor pairs represent the same effective update. We show that this redundancy makes multi-objective optimization representation-dependent. Equivalent factorizations of the same adapter can change cross-objective gradient geometry, even reversing conflict signs and altering the Pareto weights selected by gradient-based methods. Moreover, even when the same effective combined gradient is fixed, updating equivalent raw factors can produce different next effective models. Thus, representation dependence arises separately in both multi-objective decision making and update execution. We formalize representation correctness for the complete optimizer and give conditions under which equivalent LoRA representations induce identical effective trajectories. Based on this principle, we develop Quotient Pareto Adaptation (QPA), which performs Pareto optimization directly in the effective fixed-rank geometry. We further derive Stiefel Pareto Adaptation (SPA) as a complementary representation-correct construction under a distinct geometry. Both extend to transported first-moment momentum with pathwise guarantees. We evaluate QPA and SPA on helpfulness-safety Pareto frontiers across Qwen and Llama models and on external safety benchmarks.