# Understanding How Consistency Works in Federated Learning via Stage-wise Relaxed Initialization

arXiv: https://arxiv.org/abs/2306.05706 · HTML: https://mlsanity.com/paper/2306.05706/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- Builds on 45 of its 53 references that are in the map.
- First posted on arXiv 9 Jun 2023.

## Builds on (most cited first, 30 of 45)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [Communication-Efficient Learning of Deep Networks from Decentralized Data](https://mlsanity.com/paper/1602.05629/) (2016)
- [Advances and Open Problems in Federated Learning](https://mlsanity.com/paper/1912.04977/) (2019)
- [Federated Optimization in Heterogeneous Networks](https://mlsanity.com/paper/1812.06127/) (2018)
- [Adaptive Federated Optimization](https://mlsanity.com/paper/2003.00295/) (2020)
- [Tackling the Objective Inconsistency Problem in Heterogeneous Federated Optimization](https://mlsanity.com/paper/2007.07481/) (2020)
- [Measuring the Effects of Non-Identical Data Distribution for Federated Visual Classification](https://mlsanity.com/paper/1909.06335/) (2019)
- [Spectrally-normalized margin bounds for neural networks](https://mlsanity.com/paper/1706.08498/) (2017)
- [Federated Learning Based on Dynamic Regularization](https://mlsanity.com/paper/2111.04263/) (2021)
- [Train faster, generalize better: Stability of stochastic gradient descent](https://mlsanity.com/paper/1509.01240/) (2015)
- [SAGA: A Fast Incremental Gradient Method With Support for Non-Strongly Convex Composite Objectives](https://mlsanity.com/paper/1407.0202/) (2014)
- [FedProto: Federated Prototype Learning across Heterogeneous Clients](https://mlsanity.com/paper/2105.00243/) (2021)
- [A PAC-Bayesian Approach to Spectrally-Normalized Margin Bounds for Neural Networks](https://mlsanity.com/paper/1707.09564/) (2017)
- [The Non-IID Data Quagmire of Decentralized Machine Learning](https://mlsanity.com/paper/1910.00189/) (2019)
- [Don't Use Large Mini-Batches, Use Local SGD](https://mlsanity.com/paper/1808.07217/) (2018)
- [Achieving Linear Speedup with Partial Worker Participation in Non-IID Federated Learning](https://mlsanity.com/paper/2101.11203/) (2021)
- [FedDC: Federated Learning with Non-IID Data via Local Drift Decoupling and Correction](https://mlsanity.com/paper/2203.11751/) (2022)
- [Minibatch vs Local SGD for Heterogeneous Distributed Learning](https://mlsanity.com/paper/2006.04735/) (2020)
- [FedPD: A Federated Learning Framework with Optimal Rates and Adaptivity to Non-IID Data](https://mlsanity.com/paper/2005.11418/) (2020)
- [Generalized Federated Learning via Sharpness Aware Minimization](https://mlsanity.com/paper/2206.02618/) (2022)
- [SlowMo: Improving Communication-Efficient Distributed SGD with Slow Momentum](https://mlsanity.com/paper/1910.00643/) (2019)
- [Robust Federated Learning: The Case of Affine Distribution Shifts](https://mlsanity.com/paper/2006.08907/) (2020)
- [Improving Generalization in Federated Learning by Seeking Flat Minima](https://mlsanity.com/paper/2203.11834/) (2022)
- [Data-Dependent Stability of Stochastic Gradient Descent](https://mlsanity.com/paper/1703.01678/) (2017)
- [Local SGD: Unified Theory and New Efficient Methods](https://mlsanity.com/paper/2011.02828/) (2020)
- [FedCM: Federated Learning with Client-level Momentum](https://mlsanity.com/paper/2106.10874/) (2021)
- [From Local SGD to Local Fixed-Point Methods for Federated Learning](https://mlsanity.com/paper/2004.01442/) (2020)
- [Generalizable Adversarial Training via Spectral Normalization](https://mlsanity.com/paper/1811.07457/) (2018)
- [Convergence and Accuracy Trade-Offs in Federated Learning and Meta-Learning](https://mlsanity.com/paper/2103.05032/) (2021)

## Built on by (most cited first, 21 of 21)

- [Locally Estimated Global Perturbations are Better than Local Perturbations for Federated Sharpness-aware Minimization](https://mlsanity.com/paper/2405.18890/) (2024)
- [FedGPS: Statistical Rectification Against Data Heterogeneity in Federated Learning](https://mlsanity.com/paper/2510.20250/) (2025)
- [Towards Understanding Generalization and Stability Gaps between Centralized and Decentralized Federated Learning](https://mlsanity.com/paper/2310.03461/) (2023)
- [A Unified Analysis for Finite Weight Averaging](https://mlsanity.com/paper/2411.13169/) (2024)
- [Enhanced Federated Optimization: Adaptive Unbiased Client Sampling with Reduced Variance](https://mlsanity.com/paper/2310.02698/) (2023)
- [A-FedPD: Aligning Dual-Drift is All Federated Primal-Dual Learning Needs](https://mlsanity.com/paper/2409.18915/) (2024)
- [Boosting the Performance of Decentralized Federated Learning via Catalyst Acceleration](https://mlsanity.com/paper/2410.07272/) (2024)
- [FedCVD: The First Real-World Federated Learning Benchmark on Cardiovascular Disease Data](https://mlsanity.com/paper/2411.07050/) (2024)
- [Mosaic: Data-Free Knowledge Distillation via Mixture-of-Experts for Heterogeneous Distributed Environments](https://mlsanity.com/paper/2505.19699/) (2025)
- [Unveiling the Power of Multiple Gossip Steps: A Stability-Based Generalization Analysis in Decentralized Training](https://mlsanity.com/paper/2510.07980/) (2025)
- [OledFL: Unleashing the Potential of Decentralized Federated Learning via Opposite Lookahead Enhancement](https://mlsanity.com/paper/2410.06482/) (2024)
- [Neighborhood and Global Perturbations Supported SAM in Federated Learning: From Local Tweaks To Global Awareness](https://mlsanity.com/paper/2408.14144/) (2024)
- [Convergent Differential Privacy Analysis for General Federated Learning](https://mlsanity.com/paper/2408.15621/) (2024)
- [Understanding Generalization of Federated Learning: the Trade-off between Model Stability and Optimization](https://mlsanity.com/paper/2411.16303/) (2024)
- [Accelerated Training of Federated Learning via Second-Order Methods](https://mlsanity.com/paper/2505.23588/) (2025)
- [Why Federated Optimization Fails to Achieve Perfect Fitting? A Theoretical Perspective on Client-Side Optima](https://mlsanity.com/paper/2511.00469/) (2025)
- [FedRG: Unleashing the Representation Geometry for Federated Learning with Noisy Clients](https://mlsanity.com/paper/2603.19722/) (2026)
- [Rethinking the Personalized Relaxed Initialization in the Federated Learning: Consistency and Generalization](https://mlsanity.com/paper/2604.12768/) (2026)
- [Cooperate to Compete: Strategic Data Generation and Incentivization Framework for Coopetitive Cross-Silo Federated Learning](https://mlsanity.com/paper/2604.14886/) (2026)
- [Learning Theory of the SVRG: Generalization and Convergence Analysis](https://mlsanity.com/paper/2605.28513/) (2026)
- [From Data Heterogeneity to Convergence: A Data-Centric Review of Federated Learning](https://mlsanity.com/paper/2606.10595/) (2026)

## Abstract (arXiv)

Federated learning (FL) is a distributed paradigm that coordinates massive local clients to collaboratively train a global model via stage-wise local training processes on the heterogeneous dataset. Previous works have implicitly studied that FL suffers from the ``client-drift'' problem, which is caused by the inconsistent optimum across local clients. However, till now it still lacks solid theoretical analysis to explain the impact of this local inconsistency. To alleviate the negative impact of the ``client drift'' and explore its substance in FL, in this paper, we first design an efficient FL algorithm textitFedInit, which allows employing the personalized relaxed initialization state at the beginning of each local training stage. Specifically, textitFedInit initializes the local state by moving away from the current global state towards the reverse direction of the latest local state. This relaxed initialization helps to revise the local divergence and enhance the local consistency level. Moreover, to further understand how inconsistency disrupts performance in FL, we introduce the excess risk analysis and study the divergence term to investigate the test error of the proposed textitFedInit method. Our studies show that optimization error is not sensitive to this local inconsistency, while it mainly affects the generalization error bound in textitFedInit. Extensive experiments are conducted to validate this conclusion. Our proposed textitFedInit could achieve state-of-the-art~(SOTA) results compared to several advanced benchmarks without any additional costs. Meanwhile, stage-wise relaxed initialization could also be incorporated into the current advanced algorithms to achieve higher performance in the FL paradigm.