Abstract
Low-Rank Adaptation (LoRA) reparameterizes a weight update as a product of two low-rank factors, but the Jacobian JG of the generator mapping the factors to the weight matrix is rank-deficient, so the factor-space preconditioner JG∗FtJG induced by any W-space preconditioner Ft 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 JG∗FtJG to use, and (ii) which Ft on W to use. Existing methods occupy four families along these axes: factor-space adaptive updates, block-diagonal surrogates for JG∗JG, Frobenius-residual pseudoinverse methods, and Riemannian manifold constraint. Within this design space, a gradient-statistics-aware Ft 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 Ht on W and selecting from the resulting factor-space solution family the element minimizing an Ht-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 Ht-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.
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Jul 20, 2026cs.LG
Low-rank adaptation (LoRA) makes finetuning large language models cheaper by adding to each weight matrix a trainable low-rank update parameterized as the product of two matrices. These matrices are usually trained with Adam, which treats them as a single flat vector of parameters and ignores both the matrix and product structure of LoRA. Applying a matrix-aware optimizer such as Muon to each factor does not consistently improve over Adam, and neither do the product-aware Muon variants proposed in concurrent works. To realize consistent gains, we introduce PoLoRA, a Preconditioned Orthogonalized LoRA optimizer built from three ingredients: a product-aware spectral update direction, curvature preconditioning derived from controlling the per-sample loss change, and a magnitude rule that controls the sizes of both the factor and merged updates. We evaluate PoLoRA on instruction-tuning datasets for code and math across models from 1B to 8B parameters, and find that it reaches the final held-out loss achieved by tuned Adam in 1.2-1.7 times fewer steps, while adding at most 3% per-step overhead. Compared to Adam, PoLoRA is also less sensitive to the learning rate, and its optimal learning rate is stable across ranks.
Nikhil Ghosh, Tetiana Parshakova, Robert M. Gower
Sep 14, 2026cs.LG
Low-Rank Adaptation (LoRA) is an effective approach for adapting large pretrained models by learning low-rank weight updates. In practice, the LoRA rank is used to control an adapter's parameter budget and representational capacity. We show that this view is incomplete: while the nominal rank determines the representational capacity, the optimizer shapes how much of that capacity is used in the induced weight-space updates. In a case study of GPT-2 adaptation with LoRA, we observe a strong rank-dependent optimizer effect. Despite using the same nominal rank, AdamW often produces per-step updates with concentrated singular spectra and low effective rank, whereas Muon uses a richer set of directions and benefits more consistently from increasing LoRA rank. These observations motivate ISO-LoRA, an optimizer that couples the LoRA factor updates through spectral descent on the induced tangent perturbation in weight space. ISO-LoRA promotes updates that distribute energy more evenly across singular directions, improving rank utilization while preserving compatibility with the LoRA parameterization. We complement this design with theoretical guarantees showing that ISO-LoRA can achieve higher effective rank than standard factor-wise optimizers through a one-step analysis under a stylized spiked-gradient model. We validate this design on language-model adaptation across 0.1B-7B-parameter models, where ISO-LoRA improves effective rank and downstream performance, with the strongest gains at moderate-to-large LoRA ranks. Our results highlight rank utilization as a key factor in LoRA optimization and suggest that optimizer design offers an important path toward stronger parameter-efficient adaptation.
Zihan Zhu, Zhehang Du, Xuyang Chen +5
Jul 28, 2026cs.LG
Low-rank adaptation (LoRA) fine-tunes large pretrained models at a fraction of the cost of full fine-tuning, but its performance depends strongly on how the adapters are initialized. Recent schemes initialize the adapters from the downstream loss gradient: some project the raw gradient onto its top directions, while others first whiten it with an estimate of the loss curvature. We show that these seemingly distinct methods are points on a single continuum: a two-parameter family of preconditioned gradient initializations, which we call Unified LoRA (ULoRA), governed by a spectral whitening exponent and an Adam-like diagonal exponent. Sweeping this family under a full learning-rate search, we find that no single fixed preconditioning strength dominates: the best operating point is task-dependent and frequently lies strictly inside the family, away from the published endpoints. Treated as an upper bound of this family, a tuned ULoRA configuration matches or exceeds full fine-tuning on all five GLUE tasks with RoBERTa-base and is competitive with the strongest baselines on GSM8K with LLaMA-2-7B. Our deployable, search-free variant, ULoRA-Auto, selects per-layer exponents from measured spectral statistics, approaches this upper bound at no additional search cost, and ranks at or near the top among deployable LoRA methods. Our results show that a principled design space for LoRA initialization and curvature preconditioning should be treated as a tunable dimension rather than a fixed design decision.
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