# EF-Feddr: communication-efficient federated learning with Douglas–Rachford splitting and error feedback

## TL;DR

EF-Feddr is a new communication-efficient federated learning algorithm that improves training efficiency and convergence in privacy-preserving settings.

## Contribution

EF-Feddr combines relaxed Douglas–Rachford splitting with error feedback to achieve efficient convergence under biased compression.

## Key findings

- EF-Feddr achieves a convergence rate of O(1/K) and communication complexity of O(1/ε²).
- Experiments on FEMNIST, Shakespeare, and synthetic data show EF-Feddr's effectiveness in non-IID settings.
- Error feedback prevents convergence degradation caused by biased compression methods like top-k sparsification.

## Abstract

Federated learning (FL) is a distributed machine learning paradigm that preserves data privacy and mitigates data silos. Nevertheless, frequent communication between clients and the server often becomes a major bottleneck, restricting training efficiency and scalability.

To address this challenge, we propose a novel communication-efficient algorithm, EF-Feddr, for federated composite optimization, where the objective function includes a potentially non-smooth regularization term and local datasets are non-IID. Our method is built upon the relaxed Douglas–Rachford splitting method and incorporates error feedback (EF)—a widely adopted compression framework—to ensure convergence when biased compression (e.g., top-k sparsification) is applied.

Under the partial client participation setting, our theoretical analysis demonstrates that EF-Feddr achieves a fast convergence rate of O(1/K) and a communication complexity of O(1/ε2). Comprehensive experiments conducted on the FEMNIST and Shakespeare benchmarks, as well as controlled synthetic data, consistently validate the efficacy of EF-Feddr across diverse scenarios.

The results confirm that the integration of error feedback with the relaxed Douglas–Rachford splitting method in EF-Feddr effectively overcomes the convergence degradation typically caused by biased compression, thereby offering a practical and efficient solution for communication-constrained federated learning.