# Byzantine-Robust Distributed SGD: A Unified Analysis and Tight Error Bounds

arXiv: https://arxiv.org/abs/2604.10179 · HTML: https://mlsanity.com/paper/2604.10179/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 36 of its 48 references that are in the map.
- First posted on arXiv 11 Apr 2026.

## Builds on (most cited first, 30 of 36)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Communication-Efficient Learning of Deep Networks from Decentralized Data](https://mlsanity.com/paper/1602.05629/) (2016)
- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (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)
- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [Federated Learning with Non-IID Data](https://mlsanity.com/paper/1806.00582/) (2018)
- [Byzantine-Robust Distributed Learning: Towards Optimal Statistical Rates](https://mlsanity.com/paper/1803.01498/) (2018)
- [Federated Optimization: Distributed Machine Learning for On-Device Intelligence](https://mlsanity.com/paper/1610.02527/) (2016)
- [Robust Aggregation for Federated Learning](https://mlsanity.com/paper/1912.13445/) (2019)
- [A Field Guide to Federated Optimization](https://mlsanity.com/paper/2107.06917/) (2021)
- [A Little Is Enough: Circumventing Defenses For Distributed Learning](https://mlsanity.com/paper/1902.06156/) (2019)
- [RSA: Byzantine-Robust Stochastic Aggregation Methods for Distributed Learning from Heterogeneous Datasets](https://mlsanity.com/paper/1811.03761/) (2018)
- [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)
- [Learning from History for Byzantine Robust Optimization](https://mlsanity.com/paper/2012.10333/) (2020)
- [Byzantine-Robust Learning on Heterogeneous Datasets via Bucketing](https://mlsanity.com/paper/2006.09365/) (2020)
- [Stochastic Controlled Averaging for Federated Learning with Communication Compression](https://mlsanity.com/paper/2308.08165/) (2023)
- [Generalized Byzantine-tolerant SGD](https://mlsanity.com/paper/1802.10116/) (2018)
- [A Unified Theory of SGD: Variance Reduction, Sampling, Quantization and Coordinate Descent](https://mlsanity.com/paper/1905.11261/) (2019)
- [Byzantine Stochastic Gradient Descent](https://mlsanity.com/paper/1803.08917/) (2018)
- [Federated Variance-Reduced Stochastic Gradient Descent with Robustness to Byzantine Attacks](https://mlsanity.com/paper/1912.12716/) (2019)
- [Differentially Private Federated Learning on Heterogeneous Data](https://mlsanity.com/paper/2111.09278/) (2021)
- [Fixing by Mixing: A Recipe for Optimal Byzantine ML under Heterogeneity](https://mlsanity.com/paper/2302.01772/) (2023)
- [Byzantine Machine Learning Made Easy by Resilient Averaging of Momentums](https://mlsanity.com/paper/2205.12173/) (2022)
- [Byzantine-Resilient Non-Convex Stochastic Gradient Descent](https://mlsanity.com/paper/2012.14368/) (2020)
- [Momentum Benefits Non-IID Federated Learning Simply and Provably](https://mlsanity.com/paper/2306.16504/) (2023)
- [Quantized Federated Learning under Transmission Delay and Outage Constraints](https://mlsanity.com/paper/2106.09397/) (2021)
- [New Convergence Aspects of Stochastic Gradient Algorithms](https://mlsanity.com/paper/1811.12403/) (2018)
- [Byzantine-Resilient High-Dimensional Federated Learning](https://mlsanity.com/paper/2006.13041/) (2020)

## Abstract (arXiv)

Byzantine-robust distributed optimization relies on robust aggregation rules to mitigate the influence of malicious Byzantine workers. Despite the proliferation of such rules, a unified convergence analysis framework that accommodates general data heterogeneity is lacking. In this work, we provide a thorough convergence theory of Byzantine-robust distributed stochastic gradient descent (SGD), analyzing variants both with and without local momentum. We establish the convergence rates for nonconvex smooth objectives and those satisfying the Polyak-Lojasiewicz condition under a general data heterogeneity assumption. Our analysis reveals that while stochasticity and data heterogeneity introduce unavoidable error floors, local momentum provably reduces the error component induced by stochasticity. Furthermore, we derive matching lower bounds to demonstrate that the upper bounds obtained in our analysis are tight and characterize the fundamental limits of Byzantine resilience under stochasticity and data heterogeneity. Empirical results support our theoretical findings.