# An Efficient Subspace Algorithm for Federated Learning on Heterogeneous Data

arXiv: https://arxiv.org/abs/2509.05213 · HTML: https://mlsanity.com/paper/2509.05213/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 43 of its 56 references that are in the map.
- First posted on arXiv 5 Sep 2025.

## Builds on (most cited first, 30 of 43)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [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)
- [Llama 2: Open Foundation and Fine-Tuned Chat Models](https://mlsanity.com/paper/2307.09288/) (2023)
- [Pointer Sentinel Mixture Models](https://mlsanity.com/paper/1609.07843/) (2016)
- [Federated Optimization in Heterogeneous Networks](https://mlsanity.com/paper/1812.06127/) (2018)
- [Federated Learning: Strategies for Improving Communication Efficiency](https://mlsanity.com/paper/1610.05492/) (2016)
- [Federated Learning Based on Dynamic Regularization](https://mlsanity.com/paper/2111.04263/) (2021)
- [Towards Personalized Federated Learning](https://mlsanity.com/paper/2103.00710/) (2021)
- [GaLore: Memory-Efficient LLM Training by Gradient Low-Rank Projection](https://mlsanity.com/paper/2403.03507/) (2024)
- [Rethinking Machine Unlearning for Large Language Models](https://mlsanity.com/paper/2402.08787/) (2024)
- [ReLoRA: High-Rank Training Through Low-Rank Updates](https://mlsanity.com/paper/2307.05695/) (2023)
- [Improving LoRA in Privacy-preserving Federated Learning](https://mlsanity.com/paper/2403.12313/) (2024)
- [Mime: Mimicking Centralized Stochastic Algorithms in Federated Learning](https://mlsanity.com/paper/2008.03606/) (2020)
- [Stochastic Variance Reduction for Nonconvex Optimization](https://mlsanity.com/paper/1603.06160/) (2016)
- [FedPD: A Federated Learning Framework with Optimal Rates and Adaptivity to Non-IID Data](https://mlsanity.com/paper/2005.11418/) (2020)
- [Adam-mini: Use Fewer Learning Rates To Gain More](https://mlsanity.com/paper/2406.16793/) (2024)
- [Linear Convergence in Federated Learning: Tackling Client Heterogeneity and Sparse Gradients](https://mlsanity.com/paper/2102.07053/) (2021)
- [Revisiting Zeroth-Order Optimization for Memory-Efficient LLM Fine-Tuning: A Benchmark](https://mlsanity.com/paper/2402.11592/) (2024)
- [Flora: Low-Rank Adapters Are Secretly Gradient Compressors](https://mlsanity.com/paper/2402.03293/) (2024)
- [Federated Fine-tuning of Large Language Models under Heterogeneous Tasks and Client Resources](https://mlsanity.com/paper/2402.11505/) (2024)
- [LISA: Layerwise Importance Sampling for Memory-Efficient Large Language Model Fine-Tuning](https://mlsanity.com/paper/2403.17919/) (2024)
- [Chain of LoRA: Efficient Fine-tuning of Language Models via Residual Learning](https://mlsanity.com/paper/2401.04151/) (2024)
- [Variance-reduced Zeroth-Order Methods for Fine-Tuning Language Models](https://mlsanity.com/paper/2404.08080/) (2024)
- [Second-Order Fine-Tuning without Pain for LLMs:A Hessian Informed Zeroth-Order Optimizer](https://mlsanity.com/paper/2402.15173/) (2024)
- [Enhancing Zeroth-order Fine-tuning for Language Models with Low-rank Structures](https://mlsanity.com/paper/2410.07698/) (2024)
- [A Unified and Refined Convergence Analysis for Non-Convex Decentralized Learning](https://mlsanity.com/paper/2110.09993/) (2021)
- [Differentially Private Low-Rank Adaptation of Large Language Model Using Federated Learning](https://mlsanity.com/paper/2312.17493/) (2023)
- [Memory-Efficient LLM Training with Online Subspace Descent](https://mlsanity.com/paper/2408.12857/) (2024)
- [Momentum Benefits Non-IID Federated Learning Simply and Provably](https://mlsanity.com/paper/2306.16504/) (2023)

## Built on by (most cited first, 4 of 4)

- [FedSLoP: Memory-Efficient Federated Learning with Low-Rank Gradient Projection](https://mlsanity.com/paper/2604.24012/) (2026)
- [Subspace Optimization for Efficient Federated Learning under Heterogeneous Data](https://mlsanity.com/paper/2604.25467/) (2026)
- [ReasFlow: Assisting Reasoning-Centric Scientific Discovery in Applied Mathematics via a Knowledge-Based Multi-Agent System](https://mlsanity.com/paper/2607.14178/) (2026)
- [Exploiting Correlations in Federated Learning: Opportunities and Practical Limitations](https://mlsanity.com/paper/2604.14751/) (2026)

## Abstract (arXiv)

This work addresses the key challenges of applying federated learning to large-scale deep neural networks, particularly the issue of client drift due to data heterogeneity across clients and the high costs of communication, computation, and memory. We propose FedSub, an efficient subspace algorithm for federated learning on heterogeneous data. Specifically, FedSub utilizes subspace projection to guarantee local updates of each client within low-dimensional subspaces, thereby reducing communication, computation, and memory costs. Additionally, it incorporates low-dimensional dual variables to mitigate client drift. We provide convergence analysis that reveals the impact of key factors such as step size and subspace projection matrices on convergence. Experimental results demonstrate its efficiency.