PoLoRA: A Preconditioned Orthogonalized LoRA Optimizer

• URL: http://arxiv.org/abs/2607.17620v1
• Date: Mon, 20 Jul 2026 07:16:31 GMT
• Title: PoLoRA: A Preconditioned Orthogonalized LoRA Optimizer
• Abstract summary: 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.<n>PoLoRA is built from three ingredients: a product-aware update direction, preconditioning derived from controlling the per-sample loss change, and a magnitude rule that controls the sizes of both the factor and merged updates.<n>We evaluate PoLoRA on instruction-tuning for datasets 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
• Score: 15.584731292085126
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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