# Unbiased Gradient Low-Rank Projection

arXiv: https://arxiv.org/abs/2510.17802 · HTML: https://mlsanity.com/paper/2510.17802/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 68 of its 81 references that are in the map.
- First posted on arXiv 20 Oct 2025.

## Builds on (most cited first, 30 of 68)

- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Training language models to follow instructions with human feedback](https://mlsanity.com/paper/2203.02155/) (2022)
- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [Exploring the Limits of Transfer Learning with a Unified Text-to-Text Transformer](https://mlsanity.com/paper/1910.10683/) (2019)
- [LoRA: Low-Rank Adaptation of Large Language Models](https://mlsanity.com/paper/2106.09685/) (2021)
- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [Visual Instruction Tuning](https://mlsanity.com/paper/2304.08485/) (2023)
- [DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning](https://mlsanity.com/paper/2501.12948/) (2025)
- [Scalable Diffusion Models with Transformers](https://mlsanity.com/paper/2212.09748/) (2022)
- [Think you have Solved Question Answering? Try ARC, the AI2 Reasoning Challenge](https://mlsanity.com/paper/1803.05457/) (2018)
- [Mamba: Linear-Time Sequence Modeling with Selective State Spaces](https://mlsanity.com/paper/2312.00752/) (2023)
- [HellaSwag: Can a Machine Really Finish Your Sentence?](https://mlsanity.com/paper/1905.07830/) (2019)
- [FlashAttention: Fast and Memory-Efficient Exact Attention with IO-Awareness](https://mlsanity.com/paper/2205.14135/) (2022)
- [Training Compute-Optimal Large Language Models](https://mlsanity.com/paper/2203.15556/) (2022)
- [PIQA: Reasoning about Physical Commonsense in Natural Language](https://mlsanity.com/paper/1911.11641/) (2019)
- [FlashAttention-2: Faster Attention with Better Parallelism and Work Partitioning](https://mlsanity.com/paper/2307.08691/) (2023)
- [Can a Suit of Armor Conduct Electricity? A New Dataset for Open Book Question Answering](https://mlsanity.com/paper/1809.02789/) (2018)
- [Invariant Risk Minimization](https://mlsanity.com/paper/1907.02893/) (2019)
- [Sharpness-Aware Minimization for Efficiently Improving Generalization](https://mlsanity.com/paper/2010.01412/) (2020)
- [Instruction-Following Evaluation for Large Language Models](https://mlsanity.com/paper/2311.07911/) (2023)
- [OLMo: Accelerating the Science of Language Models](https://mlsanity.com/paper/2402.00838/) (2024)
- [UI-TARS: Pioneering Automated GUI Interaction with Native Agents](https://mlsanity.com/paper/2501.12326/) (2025)
- [signSGD: Compressed Optimisation for Non-Convex Problems](https://mlsanity.com/paper/1802.04434/) (2018)
- [MAmmoTH: Building Math Generalist Models through Hybrid Instruction Tuning](https://mlsanity.com/paper/2309.05653/) (2023)
- [GaLore: Memory-Efficient LLM Training by Gradient Low-Rank Projection](https://mlsanity.com/paper/2403.03507/) (2024)
- [Muon is Scalable for LLM Training](https://mlsanity.com/paper/2502.16982/) (2025)
- [Sheared LLaMA: Accelerating Language Model Pre-training via Structured Pruning](https://mlsanity.com/paper/2310.06694/) (2023)
- [DataComp-LM: In search of the next generation of training sets for language models](https://mlsanity.com/paper/2406.11794/) (2024)
- [Sparsified SGD with Memory](https://mlsanity.com/paper/1809.07599/) (2018)

## Built on by (most cited first, 7 of 7)

- [Rethinking Muon Beyond Pretraining: Spectral Failures and High-Pass Remedies for VLA and RLVR](https://mlsanity.com/paper/2605.19282/) (2026)
- [StoSignSGD: Unbiased Structural Stochasticity Fixes SignSGD for Training Large Language Models](https://mlsanity.com/paper/2604.15416/) (2026)
- [Optimizer-Model Consistency: Full Finetuning with the Same Optimizer as Pretraining Forgets Less](https://mlsanity.com/paper/2605.06654/) (2026)
- [Rethinking LoRA for Data Heterogeneous Federated Learning: Subspace and State Alignment](https://mlsanity.com/paper/2602.01746/) (2026)
- [When and Why SignSGD Outperforms SGD: A Theoretical Study Based on ℓ1-norm Lower Bounds](https://mlsanity.com/paper/2605.06615/) (2026)
- [No Subspace to Track: Non-Identifiability and Optimizer State in Low-Rank Training](https://mlsanity.com/paper/2607.05872/) (2026)
- [Memory-Efficient Subspace Optimization with Residual Correction: From Momentum SGD to Muon](https://mlsanity.com/paper/2610.19844/) (2026)

## Abstract (arXiv)

Memory-efficient optimization is critical for training increasingly large language models (LLMs). A popular strategy involves gradient low-rank projection, storing only the projected optimizer states, with GaLore being a representative example. However, a significant drawback of many such methods is their lack of convergence guarantees, as various low-rank projection approaches introduce inherent biases relative to the original optimization algorithms, which contribute to performance gaps compared to full-parameter training. Aiming to tackle this problem, this paper investigates the layerwise sampling technique for debiasing low-rank projection mechanisms. In particular, an instantiation of the paradigm gives rise to a novel and unbiased low-rank optimization method built upon GaLore's mechanism and the Muon algorithm, named GaLore Unbiased with Muon (GUM). We theoretically prove our method matches the convergence guarantees of the base Muon algorithm while preserving the memory efficiency of low-rank techniques. Empirical experiments on LLM fine-tuning and pretraining also demonstrate non-trivial improvements over GaLore and even better performance than full-parameter training. Further investigation shows that the improvement of this technique comes from a more uniform distribution of knowledge inside layers, leading to more efficient utilization of the model parameter space and better memorization.