# Decentralized SGD and Average-direction SAM are Asymptotically Equivalent

arXiv: https://arxiv.org/abs/2306.02913 · HTML: https://mlsanity.com/paper/2306.02913/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- Builds on 89 of its 143 references that are in the map.
- First posted on arXiv 5 Jun 2023.

## Builds on (most cited first, 30 of 89)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Communication-Efficient Learning of Deep Networks from Decentralized Data](https://mlsanity.com/paper/1602.05629/) (2016)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Densely Connected Convolutional Networks](https://mlsanity.com/paper/1608.06993/) (2016)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Identity Mappings in Deep Residual Networks](https://mlsanity.com/paper/1603.05027/) (2016)
- [Accurate, Large Minibatch SGD: Training ImageNet in 1 Hour](https://mlsanity.com/paper/1706.02677/) (2017)
- [Sharpness-Aware Minimization for Efficiently Improving Generalization](https://mlsanity.com/paper/2010.01412/) (2020)
- [On Large-Batch Training for Deep Learning: Generalization Gap and Sharp Minima](https://mlsanity.com/paper/1609.04836/) (2016)
- [Averaging Weights Leads to Wider Optima and Better Generalization](https://mlsanity.com/paper/1803.05407/) (2018)
- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [Visualizing the Loss Landscape of Neural Nets](https://mlsanity.com/paper/1712.09913/) (2017)
- [Adaptive Federated Optimization](https://mlsanity.com/paper/2003.00295/) (2020)
- [Deep Leakage from Gradients](https://mlsanity.com/paper/1906.08935/) (2019)
- [Tackling the Objective Inconsistency Problem in Heterogeneous Federated Optimization](https://mlsanity.com/paper/2007.07481/) (2020)
- [Inverting Gradients -- How easy is it to break privacy in federated learning?](https://mlsanity.com/paper/2003.14053/) (2020)
- [Large Batch Optimization for Deep Learning: Training BERT in 76 minutes](https://mlsanity.com/paper/1904.00962/) (2019)
- [Efficient Large-Scale Language Model Training on GPU Clusters Using Megatron-LM](https://mlsanity.com/paper/2104.04473/) (2021)
- [Cyclical Learning Rates for Training Neural Networks](https://mlsanity.com/paper/1506.01186/) (2015)
- [Exploring Generalization in Deep Learning](https://mlsanity.com/paper/1706.08947/) (2017)
- [Large Batch Training of Convolutional Networks](https://mlsanity.com/paper/1708.03888/) (2017)
- [Computing Nonvacuous Generalization Bounds for Deep (Stochastic) Neural Networks with Many More Parameters than Training Data](https://mlsanity.com/paper/1703.11008/) (2017)
- [Fantastic Generalization Measures and Where to Find Them](https://mlsanity.com/paper/1912.02178/) (2019)
- [Can Decentralized Algorithms Outperform Centralized Algorithms? A Case Study for Decentralized Parallel Stochastic Gradient Descent](https://mlsanity.com/paper/1705.09056/) (2017)
- [A Unified Theory of Decentralized SGD with Changing Topology and Local Updates](https://mlsanity.com/paper/2003.10422/) (2020)
- [Entropy-SGD: Biasing Gradient Descent Into Wide Valleys](https://mlsanity.com/paper/1611.01838/) (2016)
- [See through Gradients: Image Batch Recovery via GradInversion](https://mlsanity.com/paper/2104.07586/) (2021)
- [Train longer, generalize better: closing the generalization gap in large batch training of neural networks](https://mlsanity.com/paper/1705.08741/) (2017)
- [One weird trick for parallelizing convolutional neural networks](https://mlsanity.com/paper/1404.5997/) (2014)

## Built on by (most cited first, 15 of 15)

- [On the Duality Between Sharpness-Aware Minimization and Adversarial Training](https://mlsanity.com/paper/2402.15152/) (2024)
- [Layer-wise Linear Mode Connectivity](https://mlsanity.com/paper/2307.06966/) (2023)
- [Improving Generalization and Convergence by Enhancing Implicit Regularization](https://mlsanity.com/paper/2405.20763/) (2024)
- [Task Arithmetic Through The Lens Of One-Shot Federated Learning](https://mlsanity.com/paper/2411.18607/) (2024)
- [Lookaround Optimizer: k steps around, 1 step average](https://mlsanity.com/paper/2306.07684/) (2023)
- [DFedADMM: Dual Constraints Controlled Model Inconsistency for Decentralized Federated Learning](https://mlsanity.com/paper/2308.08290/) (2023)
- [A Quadratic Synchronization Rule for Distributed Deep Learning](https://mlsanity.com/paper/2310.14423/) (2023)
- [Towards Understanding Generalization and Stability Gaps between Centralized and Decentralized Federated Learning](https://mlsanity.com/paper/2310.03461/) (2023)
- [On the Trade-off between Flatness and Optimization in Distributed Learning](https://mlsanity.com/paper/2406.20006/) (2024)
- [DICE: Data Influence Cascade in Decentralized Learning](https://mlsanity.com/paper/2507.06931/) (2025)
- [Improved Convergence Analysis of Topology Dependence in Decentralized SGD](https://mlsanity.com/paper/2606.09154/) (2026)
- [SADDLe: Sharpness-Aware Decentralized Deep Learning with Heterogeneous Data](https://mlsanity.com/paper/2405.13961/) (2024)
- [Communication-Efficient Distributed Training for Collaborative Flat Optima Recovery in Deep Learning](https://mlsanity.com/paper/2507.20424/) (2025)
- [On the Escaping Efficiency of Distributed Adversarial Training Algorithms](https://mlsanity.com/paper/2509.11337/) (2025)
- [Decentralized SGD with Controlled Disagreement Finds Flatter Minima](https://mlsanity.com/paper/2602.02899/) (2026)

## Abstract (arXiv)

Decentralized stochastic gradient descent (D-SGD) allows collaborative learning on massive devices simultaneously without the control of a central server. However, existing theories claim that decentralization invariably undermines generalization. In this paper, we challenge the conventional belief and present a completely new perspective for understanding decentralized learning. We prove that D-SGD implicitly minimizes the loss function of an average-direction Sharpness-aware minimization (SAM) algorithm under general non-convex non-β-smooth settings. This surprising asymptotic equivalence reveals an intrinsic regularization-optimization trade-off and three advantages of decentralization: (1) there exists a free uncertainty evaluation mechanism in D-SGD to improve posterior estimation; (2) D-SGD exhibits a gradient smoothing effect; and (3) the sharpness regularization effect of D-SGD does not decrease as total batch size increases, which justifies the potential generalization benefit of D-SGD over centralized SGD (C-SGD) in large-batch scenarios. The code is available at https://github.com/Raiden-Zhu/ICML-2023-DSGD-and-SAM.