# Faster Than SVD, Smarter Than SGD: The OPLoRA Alternating Update

arXiv: https://arxiv.org/abs/2509.19977 · HTML: https://mlsanity.com/paper/2509.19977/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 20 references that are in the map.
- First posted on arXiv 24 Sep 2025.

## Builds on (most cited first, 17 of 17)

- [LoRA: Low-Rank Adaptation of Large Language Models](https://mlsanity.com/paper/2106.09685/) (2021)
- [RoBERTa: A Robustly Optimized BERT Pretraining Approach](https://mlsanity.com/paper/1907.11692/) (2019)
- [GLUE: A Multi-Task Benchmark and Analysis Platform for Natural Language Understanding](https://mlsanity.com/paper/1804.07461/) (2018)
- [Optimizing Neural Networks with Kronecker-factored Approximate Curvature](https://mlsanity.com/paper/1503.05671/) (2015)
- [DoRA: Weight-Decomposed Low-Rank Adaptation](https://mlsanity.com/paper/2402.09353/) (2024)
- [GaLore: Memory-Efficient LLM Training by Gradient Low-Rank Projection](https://mlsanity.com/paper/2403.03507/) (2024)
- [Shampoo: Preconditioned Stochastic Tensor Optimization](https://mlsanity.com/paper/1802.09568/) (2018)
- [AdaLoRA: Adaptive Budget Allocation for Parameter-Efficient Fine-Tuning](https://mlsanity.com/paper/2303.10512/) (2023)
- [ReLoRA: High-Rank Training Through Low-Rank Updates](https://mlsanity.com/paper/2307.05695/) (2023)
- [Accelerating Ill-Conditioned Low-Rank Matrix Estimation via Scaled Gradient Descent](https://mlsanity.com/paper/2005.08898/) (2020)
- [Riemannian Preconditioned LoRA for Fine-Tuning Foundation Models](https://mlsanity.com/paper/2402.02347/) (2024)
- [Understanding AdamW through Proximal Methods and Scale-Freeness](https://mlsanity.com/paper/2202.00089/) (2022)
- [The Power of Preconditioning in Overparameterized Low-Rank Matrix Sensing](https://mlsanity.com/paper/2302.01186/) (2023)
- [Preconditioned Gradient Descent for Overparameterized Nonconvex Burer--Monteiro Factorization with Global Optimality Certification](https://mlsanity.com/paper/2206.03345/) (2022)
- [LoFT: Low-Rank Adaptation That Behaves Like Full Fine-Tuning](https://mlsanity.com/paper/2505.21289/) (2025)
- [Preconditioned Gradient Descent for Over-Parameterized Nonconvex Matrix Factorization](https://mlsanity.com/paper/2504.09708/) (2025)
- [Low-rank Momentum Factorization for Memory Efficient Training](https://mlsanity.com/paper/2507.08091/) (2025)

## Built on by (most cited first, 1 of 1)

- [ODELoRA: Training Low-Rank Adaptation by Solving Ordinary Differential Equations](https://mlsanity.com/paper/2602.07479/) (2026)

## Abstract (arXiv)

Low-Rank Adaptation (LoRA) fine-tunes large models by learning low-rank updates on top of frozen weights, dramatically reducing trainable parameters and memory. However, there is still a gap between full training with low-rank projections (SVDLoRA) and LoRA fine-tuning, indicating that LoRA steps can be further improved. In this study, we propose OPLoRA, a memory-efficient optimizer that closes this gap by casting LoRA optimization as an interpretable sub-problem and solving it efficiently with alternating least squares updates, where 1-2 alternating steps are empirically found to be sufficient to closely match truncated SVD without ever forming the full matrix. We also retrieve the recently proposed preconditioning methods for LoRA as a special case. OPLoRA supports momentum by maintaining a low-rank estimate using the same subroutine (LoRSum) for computing the step, with a memory budget of 3 times the number of LoRA parameters (i.e., same as Adam). We also propose an experimental scaled variant that uses the K-FAC metric, which could be of interest. Across a linear task, MNIST, CIFAR-100, and RoBERTa-base (MNLI), OPLoRA consistently approaches SVDLoRA's performance using significantly less memory.