# Subspace Optimization for Large Language Models with Convergence Guarantees

arXiv: https://arxiv.org/abs/2410.11289 · HTML: https://mlsanity.com/paper/2410.11289/

## Key facts

- Cited by 34 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Oct 2024.

## Built on by (most cited first, 30 of 34)

- [APOLLO: SGD-like Memory, AdamW-level Performance](https://mlsanity.com/paper/2412.05270/) (2024)
- [SubTrack++ : Gradient Subspace Tracking for Scalable LLM Training](https://mlsanity.com/paper/2502.01586/) (2025)
- [Powering Up Zeroth-Order Training via Subspace Gradient Orthogonalization](https://mlsanity.com/paper/2602.17155/) (2026)
- [Greedy Low-Rank Gradient Compression for Distributed Learning with Convergence Guarantees](https://mlsanity.com/paper/2507.08784/) (2025)
- [On the Convergence Rate of LoRA Gradient Descent](https://mlsanity.com/paper/2512.18248/) (2025)
- [AltLoRA: Towards Better Gradient Approximation in Low-Rank Adaptation with Alternating Projections](https://mlsanity.com/paper/2505.12455/) (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)
- [CE-LoRA: Computation-Efficient LoRA Fine-Tuning for Language Models](https://mlsanity.com/paper/2502.01378/) (2025)
- [Scalable Parameter and Memory Efficient Pretraining for LLM: Recent Algorithmic Advances and Benchmarking](https://mlsanity.com/paper/2505.22922/) (2025)
- [An Efficient Subspace Algorithm for Federated Learning on Heterogeneous Data](https://mlsanity.com/paper/2509.05213/) (2025)
- [AdaPM: a Partial Momentum Algorithm for LLM Training](https://mlsanity.com/paper/2510.09103/) (2025)
- [Memory-Efficient Differentially Private Training with Gradient Random Projection](https://mlsanity.com/paper/2506.15588/) (2025)
- [Rethinking LoRA for Data Heterogeneous Federated Learning: Subspace and State Alignment](https://mlsanity.com/paper/2602.01746/) (2026)
- [FedKRSO: Communication and Memory Efficient Federated Fine-Tuning of Large Language Models](https://mlsanity.com/paper/2602.03019/) (2026)
- [GWT: Scalable Optimizer State Compression for Large Language Model Training](https://mlsanity.com/paper/2501.07237/) (2025)
- [MLorc: Momentum Low-rank Compression for Memory Efficient Large Language Model Adaptation](https://mlsanity.com/paper/2506.01897/) (2025)
- [Train Large, Deploy Compact: Structured Compression for Compact Low-Rank Adaptation](https://mlsanity.com/paper/2510.00192/) (2025)
- [Geometrically Principled Randomized Optimization for Efficient LLM Training](https://mlsanity.com/paper/2510.01878/) (2025)
- [Convergence Analysis of Randomized Subspace Normalized SGD under Heavy-Tailed Noise](https://mlsanity.com/paper/2601.20399/) (2026)
- [PRAC: Principal-Random Subspace for LLM Activation Compression and Memory-Efficient Training](https://mlsanity.com/paper/2602.23111/) (2026)
- [Omni-Masked Gradient Descent: Memory-Efficient Optimization via Mask Traversal with Improved Convergence](https://mlsanity.com/paper/2603.05960/) (2026)
- [Optimal low-rank stochastic gradient estimation for LLM training](https://mlsanity.com/paper/2603.20632/) (2026)
- [FedSLoP: Memory-Efficient Federated Learning with Low-Rank Gradient Projection](https://mlsanity.com/paper/2604.24012/) (2026)
- [Pro-KLShampoo: Projected KL-Shampoo with Whitening Recovered by Orthogonalization](https://mlsanity.com/paper/2605.06316/) (2026)
- [From PowerSGD to PowerSGD+: Low-Rank Gradient Compression for Distributed Optimization with Convergence Guarantees](https://mlsanity.com/paper/2509.11254/) (2025)
- [An All-Reduce Compatible Top-K Compressor for Communication-Efficient Distributed Learning](https://mlsanity.com/paper/2510.26709/) (2025)
- [From O(mn) to O(r2): Two-Sided Low-Rank Communication for Adam in Distributed Training with Memory Efficiency](https://mlsanity.com/paper/2602.08007/) (2026)
- [BROS: Bias-Corrected Randomized Subspaces for Memory-Efficient Single-Loop Bilevel Optimization](https://mlsanity.com/paper/2605.10288/) (2026)

## Abstract (arXiv)

Subspace optimization algorithms, such as GaLore (Zhao et al., 2024), have gained attention for pre-training and fine-tuning large language models (LLMs) due to their memory efficiency. However, their convergence guarantees remain unclear, particularly in stochastic settings. In this paper, we reveal that GaLore does not always converge to the optimal solution and provide an explicit counterexample to support this finding. We further explore the conditions under which GaLore achieves convergence, showing that it does so when either (i) a sufficiently large mini-batch size is used or (ii) the gradient noise is isotropic. More significantly, we introduce GoLore (Gradient random Low-rank projection), a novel variant of GaLore that provably converges in typical stochastic settings, even with standard batch sizes. Our convergence analysis extends naturally to other subspace optimization algorithms. Finally, we empirically validate our theoretical results and thoroughly test the proposed mechanisms. Codes are available at https://github.com/pkumelon/Golore.