AdaPreLoRA: Adafactor Preconditioned Low-Rank Adaptation

Low-Rank Adaptation (LoRA) reparameterizes a weight update as a product of two low-rank factors, but the Jacobian $J\_\mathcal{G}$ of the generator mapping the factors to the weight matrix is rank-deficient, so the factor-space preconditioner $J\_\mathcal{G}^\* \mathcal{F}\_t J\_\mathcal{G}$ induced by any ${W}$-space preconditioner $\mathcal{F}\_t$ 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 $J\_\mathcal{G}^\* \mathcal{F}\_t J\_\mathcal{G}$ to use, and (ii) which $\mathcal{F}\_t$ on ${W}$ to use. Existing methods occupy four families along these axes: factor-space adaptive updates, block-diagonal surrogates for $J\_\mathcal{G}^\* J\_\mathcal{G}$, Frobenius-residual pseudoinverse methods, and Riemannian manifold constraint. Within this design space, a gradient-statistics-aware $\mathcal{F}\_t$ paired with a closed-form factor-space solve at $\mathcal{O}((m+n)r)$ memory remains underexplored. We propose AdaPreLoRA, which fills this gap by adopting the Adafactor diagonal Kronecker preconditioner $\mathcal{H}\_t$ on ${W}$ and selecting from the resulting factor-space solution family the element minimizing an $\mathcal{H}\_t$-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 $\mathcal{H}\_t$-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.