# From PowerSGD to PowerSGD+: Low-Rank Gradient Compression for Distributed Optimization with Convergence Guarantees

arXiv: https://arxiv.org/abs/2509.11254 · HTML: https://mlsanity.com/paper/2509.11254/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 25 references that are in the map.
- First posted on arXiv 14 Sep 2025.

## Builds on (most cited first, 18 of 18)

- [LoRA: Low-Rank Adaptation of Large Language Models](https://mlsanity.com/paper/2106.09685/) (2021)
- [Zero-Shot Text-to-Image Generation](https://mlsanity.com/paper/2102.12092/) (2021)
- [Federated Learning: Strategies for Improving Communication Efficiency](https://mlsanity.com/paper/1610.05492/) (2016)
- [GaLore: Memory-Efficient LLM Training by Gradient Low-Rank Projection](https://mlsanity.com/paper/2403.03507/) (2024)
- [Sparsified SGD with Memory](https://mlsanity.com/paper/1809.07599/) (2018)
- [Error Feedback Fixes SignSGD and other Gradient Compression Schemes](https://mlsanity.com/paper/1901.09847/) (2019)
- [TernGrad: Ternary Gradients to Reduce Communication in Distributed Deep Learning](https://mlsanity.com/paper/1705.07878/) (2017)
- [PowerSGD: Practical Low-Rank Gradient Compression for Distributed Optimization](https://mlsanity.com/paper/1905.13727/) (2019)
- [Gradient Sparsification for Communication-Efficient Distributed Optimization](https://mlsanity.com/paper/1710.09854/) (2017)
- [ATOMO: Communication-efficient Learning via Atomic Sparsification](https://mlsanity.com/paper/1806.04090/) (2018)
- [EF21: A New, Simpler, Theoretically Better, and Practically Faster Error Feedback](https://mlsanity.com/paper/2106.05203/) (2021)
- [Enhancing Zeroth-order Fine-tuning for Language Models with Low-rank Structures](https://mlsanity.com/paper/2410.07698/) (2024)
- [SLTrain: a sparse plus low-rank approach for parameter and memory efficient pretraining](https://mlsanity.com/paper/2406.02214/) (2024)
- [Subspace Optimization for Large Language Models with Convergence Guarantees](https://mlsanity.com/paper/2410.11289/) (2024)
- [Lower Bounds and Nearly Optimal Algorithms in Distributed Learning with Communication Compression](https://mlsanity.com/paper/2206.03665/) (2022)
- [Momentum Provably Improves Error Feedback!](https://mlsanity.com/paper/2305.15155/) (2023)
- [Greedy Low-Rank Gradient Compression for Distributed Learning with Convergence Guarantees](https://mlsanity.com/paper/2507.08784/) (2025)
- [Angel-PTM: A Scalable and Economical Large-scale Pre-training System in Tencent](https://mlsanity.com/paper/2303.02868/) (2023)

## Abstract (arXiv)

Low-rank gradient compression methods, such as PowerSGD, have gained attention in communication-efficient distributed optimization. However, the convergence guarantees of PowerSGD remain unclear, particularly in stochastic settings. In this paper, we show that PowerSGD does not always converge to the optimal solution and provide a clear counterexample to support this finding. To address this, we introduce PowerSGD+, which periodically updates the projection subspace via singular value decomposition, ensuring that it remains aligned with the optimal subspace. We prove that PowerSGD+ converges under standard assumptions and validate its effectiveness through empirical evaluation on large language model tasks.