Ziyun Liu, Fengmiao Bian, Jian-Feng Cai
Low-Rank Adaptation (LoRA) reparameterizes a weight update as a product of two low-rank factors, but the Jacobian $J_{G}$ of the generator mapping the factors to the weight matrix is rank-deficient, so the factor-space preconditioner $J_{G}^* {F}_t J_{G}$ induced by any ${W}$-space preconditioner ${F}_t$ is singular, and consequently the standard chain rule cannot be uniquely inverted to map a preconditioned ${W}$-space direction back to a factor-space update. We cast existing LoRA optimizers in a unified framework parameterized by two choices: (i) which invertible surrogate for $J_{G}^* {F}_t J_{G}$ to use, and (ii) which ${F}_t$ on ${W}$ to use. Existing methods occupy four families along these axes: factor-space adaptive updates, block-diagonal surrogates for $J_{G}^* J_{G}$, Frobenius-residual pseudoinverse methods, and Riemannian manifold constraint. Within this design space, a gradient-statistics-aware ${F}_t$ paired with a closed-form factor-space solve at ${O}((m+n)r)$ memory remains underexplored. We propose \textbf{AdaPreLoRA}, which fills this gap by adopting the Adafactor diagonal Kronecker preconditioner ${H}_t$ on ${W}$ and selecting from the resulting factor-space solution family the element minimizing an ${H}_t$-weighted imbalance between the two factor contributions; by construction, the resulting factor update is the closest LoRA approximation to the preconditioned ${W}$-space direction under the ${H}_t$-weighted norm. Across GPT-2 (E2E), Mistral-7B and Qwen2-7B (GLUE, ARC, GSM8K), and diffusion-model personalization, AdaPreLoRA is competitive with or improves over a representative set of LoRA optimizers while keeping peak GPU memory at the LoRA optimizer level.