# Closed-form merging of parameter-efficient modules for Federated Continual Learning

arXiv: https://arxiv.org/abs/2410.17961 · HTML: https://mlsanity.com/paper/2410.17961/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- Builds on 49 of its 67 references that are in the map.
- First posted on arXiv 23 Oct 2024.

## Builds on (most cited first, 30 of 49)

- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Exploring the Limits of Transfer Learning with a Unified Text-to-Text Transformer](https://mlsanity.com/paper/1910.10683/) (2019)
- [Communication-Efficient Learning of Deep Networks from Decentralized Data](https://mlsanity.com/paper/1602.05629/) (2016)
- [Overcoming catastrophic forgetting in neural networks](https://mlsanity.com/paper/1612.00796/) (2016)
- [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)
- [The Power of Scale for Parameter-Efficient Prompt Tuning](https://mlsanity.com/paper/2104.08691/) (2021)
- [Learning without Forgetting](https://mlsanity.com/paper/1606.09282/) (2016)
- [Learning to Prompt for Vision-Language Models](https://mlsanity.com/paper/2109.01134/) (2021)
- [The Many Faces of Robustness: A Critical Analysis of Out-of-Distribution Generalization](https://mlsanity.com/paper/2006.16241/) (2020)
- [Natural Adversarial Examples](https://mlsanity.com/paper/1907.07174/) (2019)
- [Editing Models with Task Arithmetic](https://mlsanity.com/paper/2212.04089/) (2022)
- [Federated Learning with Non-IID Data](https://mlsanity.com/paper/1806.00582/) (2018)
- [Learning to Prompt for Continual Learning](https://mlsanity.com/paper/2112.08654/) (2021)
- [Dark Experience for General Continual Learning: a Strong, Simple Baseline](https://mlsanity.com/paper/2004.07211/) (2020)
- [Few-Shot Parameter-Efficient Fine-Tuning is Better and Cheaper than In-Context Learning](https://mlsanity.com/paper/2205.05638/) (2022)
- [TIES-Merging: Resolving Interference When Merging Models](https://mlsanity.com/paper/2306.01708/) (2023)
- [Big Transfer (BiT): General Visual Representation Learning](https://mlsanity.com/paper/1912.11370/) (2019)
- [DualPrompt: Complementary Prompting for Rehearsal-free Continual Learning](https://mlsanity.com/paper/2204.04799/) (2022)
- [Merging Models with Fisher-Weighted Averaging](https://mlsanity.com/paper/2111.09832/) (2021)
- [ImageNet-21K Pretraining for the Masses](https://mlsanity.com/paper/2104.10972/) (2021)
- [Federated Learning on Non-IID Data Silos: An Experimental Study](https://mlsanity.com/paper/2102.02079/) (2021)
- [Do Better ImageNet Models Transfer Better?](https://mlsanity.com/paper/1805.08974/) (2018)
- [CODA-Prompt: COntinual Decomposed Attention-based Prompting for Rehearsal-Free Continual Learning](https://mlsanity.com/paper/2211.13218/) (2022)
- [Bayesian Nonparametric Federated Learning of Neural Networks](https://mlsanity.com/paper/1905.12022/) (2019)
- [Dataless Knowledge Fusion by Merging Weights of Language Models](https://mlsanity.com/paper/2212.09849/) (2022)
- [An Evaluation Dataset for Intent Classification and Out-of-Scope Prediction](https://mlsanity.com/paper/1909.02027/) (2019)
- [LoraHub: Efficient Cross-Task Generalization via Dynamic LoRA Composition](https://mlsanity.com/paper/2307.13269/) (2023)
- [FedProto: Federated Prototype Learning across Heterogeneous Clients](https://mlsanity.com/paper/2105.00243/) (2021)
- [AdaMerging: Adaptive Model Merging for Multi-Task Learning](https://mlsanity.com/paper/2310.02575/) (2023)

## Built on by (most cited first, 22 of 22)

- [Unleashing the Power of Continual Learning on Non-Centralized Devices: A Survey](https://mlsanity.com/paper/2412.13840/) (2024)
- [Merge before Forget: A Single LoRA Continual Learning via Continual Merging](https://mlsanity.com/paper/2512.23017/) (2025)
- [FedMerge: Federated Personalization via Model Merging](https://mlsanity.com/paper/2504.06768/) (2025)
- [HiLoRA: Adaptive Hierarchical LoRA Routing for Training-Free Domain Generalization](https://mlsanity.com/paper/2510.12266/) (2025)
- [DitHub: A Modular Framework for Incremental Open-Vocabulary Object Detection](https://mlsanity.com/paper/2503.09271/) (2025)
- [C^2Prompt: Class-aware Client Knowledge Interaction for Federated Continual Learning](https://mlsanity.com/paper/2509.19674/) (2025)
- [Model Merging via Multi-Teacher Knowledge Distillation](https://mlsanity.com/paper/2512.21288/) (2025)
- [Task-agnostic Low-rank Residual Adaptation for Efficient Federated Continual Fine-Tuning](https://mlsanity.com/paper/2505.12318/) (2025)
- [FedRPCA: Enhancing Federated LoRA Aggregation Using Robust PCA](https://mlsanity.com/paper/2506.01194/) (2025)
- [Task-Agnostic Federated Continual Learning via Replay-Free Gradient Projection](https://mlsanity.com/paper/2509.21606/) (2025)
- [From Parameters to Behaviors: A Survey of Model Fusion for Large Language Models](https://mlsanity.com/paper/2609.19553/) (2026)
- [FroM: Frobenius Norm-Based Data-Free Adaptive Model Merging](https://mlsanity.com/paper/2506.02478/) (2025)
- [Intrinsic Training Signals for Federated Learning Aggregation](https://mlsanity.com/paper/2507.06813/) (2025)
- [Large-Small Model Collaborative Framework for Federated Continual Learning](https://mlsanity.com/paper/2508.09489/) (2025)
- [BRAIN: Bias-Mitigation Continual Learning Approach to Vision-Brain Understanding](https://mlsanity.com/paper/2508.18187/) (2025)
- [DOLFIN: Balancing Stability and Plasticity in Federated Continual Learning](https://mlsanity.com/paper/2510.13567/) (2025)
- [Towards Robust Knowledge Removal in Federated Learning with High Data Heterogeneity](https://mlsanity.com/paper/2510.13606/) (2025)
- [HERO: A Heterogeneity-Aware Benchmark Library for Federated Continual Learning](https://mlsanity.com/paper/2607.08784/) (2026)
- [SUM: Unified Geometric Surgery on Spatio-Temporal Adaptation Vectors for Federated Class Incremental Learning](https://mlsanity.com/paper/2607.19384/) (2026)
- [Federated Continual Learning as a Distributed Drift-Plus-Penalty Control Problem](https://mlsanity.com/paper/2608.21539/) (2026)
- [What Should Federated LoRA Share? FedSAIL via Input-aware Subspace Alignment](https://mlsanity.com/paper/2609.32485/) (2026)
- [ChainLoRA: Geometry-Preserving Task Vector Merging for Continual Learning in LLMs](https://mlsanity.com/paper/2610.00431/) (2026)

## Abstract (arXiv)

Model merging has emerged as a crucial technique in Deep Learning, enabling the integration of multiple models into a unified system while preserving perfor-mance and scalability. In this respect, the compositional properties of low-rank adaptation techniques (e.g., LoRA) have proven beneficial, as simple averaging LoRA modules yields a single model that mostly integrates the capabilities of all individual modules. Building on LoRA, we take a step further by imposing that the merged model matches the responses of all learned modules. Solving this objective in closed form yields an indeterminate system with A and B as unknown variables, indicating the existence of infinitely many closed-form solutions. To address this challenge, we introduce LoRM, an alternating optimization strategy that trains one LoRA matrix at a time. This allows solving for each unknown variable individually, thus finding a unique solution. We apply our proposed methodology to Federated Class-Incremental Learning (FCIL), ensuring alignment of model responses both between clients and across tasks. Our method demonstrates state-of-the-art performance across a range of FCIL scenarios. The code to reproduce our experiments is available at github.com/aimagelab/fed-mammoth.