# Fira: Can We Achieve Full-rank Training of LLMs Under Low-rank Constraint?

arXiv: https://arxiv.org/abs/2410.01623 · HTML: https://mlsanity.com/paper/2410.01623/

## Key facts

- Cited by 43 papers in the map (data through 14 Oct 2026).
- Builds on 35 of its 50 references that are in the map.
- First posted on arXiv 2 Oct 2024.

## Builds on (most cited first, 30 of 35)

- [GPT-4 Technical Report](https://mlsanity.com/paper/2303.08774/) (2023)
- [Exploring the Limits of Transfer Learning with a Unified Text-to-Text Transformer](https://mlsanity.com/paper/1910.10683/) (2019)
- [Llama 2: Open Foundation and Fine-Tuned Chat Models](https://mlsanity.com/paper/2307.09288/) (2023)
- [Parameter-Efficient Transfer Learning for NLP](https://mlsanity.com/paper/1902.00751/) (2019)
- [Prefix-Tuning: Optimizing Continuous Prompts for Generation](https://mlsanity.com/paper/2101.00190/) (2021)
- [QLoRA: Efficient Finetuning of Quantized LLMs](https://mlsanity.com/paper/2305.14314/) (2023)
- [GLU Variants Improve Transformer](https://mlsanity.com/paper/2002.05202/) (2020)
- [Root Mean Square Layer Normalization](https://mlsanity.com/paper/1910.07467/) (2019)
- [Towards a Unified View of Parameter-Efficient Transfer Learning](https://mlsanity.com/paper/2110.04366/) (2021)
- [Training Deep Nets with Sublinear Memory Cost](https://mlsanity.com/paper/1604.06174/) (2016)
- [GLM-130B: An Open Bilingual Pre-trained Model](https://mlsanity.com/paper/2210.02414/) (2022)
- [On the Variance of the Adaptive Learning Rate and Beyond](https://mlsanity.com/paper/1908.03265/) (2019)
- [DriveLM: Driving with Graph Visual Question Answering](https://mlsanity.com/paper/2312.14150/) (2023)
- [GaLore: Memory-Efficient LLM Training by Gradient Low-Rank Projection](https://mlsanity.com/paper/2403.03507/) (2024)
- [8-bit Optimizers via Block-wise Quantization](https://mlsanity.com/paper/2110.02861/) (2021)
- [AdaLoRA: Adaptive Budget Allocation for Parameter-Efficient Fine-Tuning](https://mlsanity.com/paper/2303.10512/) (2023)
- [Why gradient clipping accelerates training: A theoretical justification for adaptivity](https://mlsanity.com/paper/1905.11881/) (2019)
- [ZeRO-Offload: Democratizing Billion-Scale Model Training](https://mlsanity.com/paper/2101.06840/) (2021)
- [LLM-Adapters: An Adapter Family for Parameter-Efficient Fine-Tuning of Large Language Models](https://mlsanity.com/paper/2304.01933/) (2023)
- [DyLoRA: Parameter Efficient Tuning of Pre-trained Models using Dynamic Search-Free Low-Rank Adaptation](https://mlsanity.com/paper/2210.07558/) (2022)
- [ReLoRA: High-Rank Training Through Low-Rank Updates](https://mlsanity.com/paper/2307.05695/) (2023)
- [The Shattered Gradients Problem: If resnets are the answer, then what is the question?](https://mlsanity.com/paper/1702.08591/) (2017)
- [Adam-mini: Use Fewer Learning Rates To Gain More](https://mlsanity.com/paper/2406.16793/) (2024)
- [Towards Theoretically Understanding Why SGD Generalizes Better Than ADAM in Deep Learning](https://mlsanity.com/paper/2010.05627/) (2020)
- [Flora: Low-Rank Adapters Are Secretly Gradient Compressors](https://mlsanity.com/paper/2402.03293/) (2024)
- [What Language Model to Train if You Have One Million GPU Hours?](https://mlsanity.com/paper/2210.15424/) (2022)
- [Chain of LoRA: Efficient Fine-tuning of Language Models via Residual Learning](https://mlsanity.com/paper/2401.04151/) (2024)
- [Fast low-rank estimation by projected gradient descent: General statistical and algorithmic guarantees](https://mlsanity.com/paper/1509.03025/) (2015)
- [Deconstructing What Makes a Good Optimizer for Language Models](https://mlsanity.com/paper/2407.07972/) (2024)
- [Batched Low-Rank Adaptation of Foundation Models](https://mlsanity.com/paper/2312.05677/) (2023)

## Built on by (most cited first, 30 of 43)

- [APOLLO: SGD-like Memory, AdamW-level Performance](https://mlsanity.com/paper/2412.05270/) (2024)
- [CoLA: Compute-Efficient Pre-Training of LLMs via Low-Rank Activation](https://mlsanity.com/paper/2502.10940/) (2025)
- [SWAN: SGD with Normalization and Whitening Enables Stateless LLM Training](https://mlsanity.com/paper/2412.13148/) (2024)
- [PoLAR: Polar-Decomposed Low-Rank Adapter Representation](https://mlsanity.com/paper/2506.03133/) (2025)
- [SUMO: Subspace-Aware Moment-Orthogonalization for Accelerating Memory-Efficient LLM Training](https://mlsanity.com/paper/2505.24749/) (2025)
- [SubTrack++ : Gradient Subspace Tracking for Scalable LLM Training](https://mlsanity.com/paper/2502.01586/) (2025)
- [Memory-Efficient LLM Pretraining via Minimalist Optimizer Design](https://mlsanity.com/paper/2506.16659/) (2025)
- [Greedy Low-Rank Gradient Compression for Distributed Learning with Convergence Guarantees](https://mlsanity.com/paper/2507.08784/) (2025)
- [FRUGAL: Memory-Efficient Optimization by Reducing State Overhead for Scalable Training](https://mlsanity.com/paper/2411.07837/) (2024)
- [LOST: Low-rank and Sparse Pre-training for Large Language Models](https://mlsanity.com/paper/2508.02668/) (2025)
- [Gradient Multi-Normalization for Stateless and Scalable LLM Training](https://mlsanity.com/paper/2502.06742/) (2025)
- [Reparameterized LLM Training via Orthogonal Equivalence Transformation](https://mlsanity.com/paper/2506.08001/) (2025)
- [GradPower: Powering Gradients for Faster Language Model Pre-Training](https://mlsanity.com/paper/2505.24275/) (2025)
- [Breaking the Frozen Subspace: Importance Sampling for Low-Rank Optimization in LLM Pretraining](https://mlsanity.com/paper/2502.05790/) (2025)
- [CR-Net: Scaling Parameter-Efficient Training with Cross-Layer Low-Rank Structure](https://mlsanity.com/paper/2509.18993/) (2025)
- [Unbiased Gradient Low-Rank Projection](https://mlsanity.com/paper/2510.17802/) (2025)
- [Stabilizing Native Low-Rank LLM Pretraining](https://mlsanity.com/paper/2602.12429/) (2026)
- [LORENZA: Enhancing Generalization in Low-Rank Gradient LLM Training via Efficient Zeroth-Order Adaptive SAM](https://mlsanity.com/paper/2502.19571/) (2025)
- [Scalable Parameter and Memory Efficient Pretraining for LLM: Recent Algorithmic Advances and Benchmarking](https://mlsanity.com/paper/2505.22922/) (2025)
- [Conda: Column-Normalized Adam for Training Large Language Models Faster](https://mlsanity.com/paper/2509.24218/) (2025)
- [POET-X: Memory-efficient LLM Training by Scaling Orthogonal Transformation](https://mlsanity.com/paper/2603.05500/) (2026)
- [FFT-based Dynamic Subspace Selection for Low-Rank Adaptive Optimization of Large Language Models](https://mlsanity.com/paper/2505.17967/) (2025)
- [Taming LLMs by Scaling Learning Rates with Gradient Grouping](https://mlsanity.com/paper/2506.01049/) (2025)
- [Train Less, Infer Faster: Efficient Model Finetuning and Compression via Structured Sparsity](https://mlsanity.com/paper/2602.09169/) (2026)
- [Taming Momentum: Rethinking Optimizer States Through Low-Rank Approximation](https://mlsanity.com/paper/2602.24283/) (2026)
- [GWT: Scalable Optimizer State Compression for Large Language Model Training](https://mlsanity.com/paper/2501.07237/) (2025)
- [Sparsity Outperforms Low-Rank Projections in Few-Shot Adaptation](https://mlsanity.com/paper/2504.12436/) (2025)
- [Memory-Efficient LLM Training by Various-Grained Low-Rank Projection of Gradients](https://mlsanity.com/paper/2505.01744/) (2025)
- [MLorc: Momentum Low-rank Compression for Memory Efficient Large Language Model Adaptation](https://mlsanity.com/paper/2506.01897/) (2025)
- [Geometrically Principled Randomized Optimization for Efficient LLM Training](https://mlsanity.com/paper/2510.01878/) (2025)

## Abstract (arXiv)

Low-rank training has emerged as a promising approach for reducing memory usage in training Large Language Models (LLMs). Previous methods either rely on decomposing weight matrices (e.g., LoRA), or seek to decompose gradient matrices (e.g., GaLore) to ensure reduced memory consumption. However, both of them constrain the training in a low-rank subspace, thus inevitably leading to sub-optimal performance. This raises a question: whether it is possible to consistently preserve the low-rank constraint for memory efficiency, while achieving full-rank training (i.e., training with full-rank gradients of full-rank weights) to avoid inferior outcomes? In this paper, we propose a new plug-and-play training framework for LLMs called Fira, as the first attempt to achieve this goal. First, we observe an interesting phenomenon during LLM training: the scaling impact of adaptive optimizers (e.g., Adam) on the gradient norm remains similar from low-rank to full-rank training. Based on this observation, we propose a norm-based scaling method, which utilizes the scaling impact of low-rank optimizers as substitutes for that of original full-rank optimizers to enable full-rank training. In this way, we can preserve the low-rank constraint in the optimizer while achieving full-rank training for better performance. Moreover, we find that there are sudden gradient rises during the optimization process, potentially causing loss spikes. To address this, we further put forward a norm-growth limiter to smooth the gradient via regulating the relative increase of gradient norms. Extensive experiments on the pre-training and fine-tuning of LLMs show that Fira outperforms both LoRA and GaLore, achieving performance that is comparable to or even better than full-rank training.