# On the Geometric Convergence of Byzantine-Resilient Distributed Optimization Algorithms

arXiv: https://arxiv.org/abs/2305.10810 · HTML: https://mlsanity.com/paper/2305.10810/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 May 2023.

## Built on by (most cited first, 6 of 6)

- [Just One Byte (per gradient): A Note on Low-Bandwidth Decentralized Language Model Finetuning Using Shared Randomness](https://mlsanity.com/paper/2306.10015/) (2023)
- [Byzantine-Robust Decentralized Stochastic Optimization with Stochastic Gradient Noise-Independent Learning Error](https://mlsanity.com/paper/2308.05292/) (2023)
- [Byzantine-Robust Gossip: Insights from a Dual Approach](https://mlsanity.com/paper/2405.03449/) (2024)
- [Generalization Error Analysis for Attack-Free and Byzantine-Resilient Decentralized Learning with Data Heterogeneity](https://mlsanity.com/paper/2506.09438/) (2025)
- [Adaptive Decentralized Federated Learning for Robust Optimization](https://mlsanity.com/paper/2512.02852/) (2025)
- [Convergence of Byzantine-Resilient Gradient Tracking via Probabilistic Edge Dropout](https://mlsanity.com/paper/2604.00449/) (2026)

## Abstract (arXiv)

The problem of designing distributed optimization algorithms that are resilient to Byzantine adversaries has received significant attention. For the Byzantine-resilient distributed optimization problem, the goal is to (approximately) minimize the average of the local cost functions held by the regular (non adversarial) agents in the network. In this paper, we provide a general algorithmic framework for Byzantine-resilient distributed optimization which includes some state-of-the-art algorithms as special cases. We analyze the convergence of algorithms within the framework, and derive a geometric rate of convergence of all regular agents to a ball around the optimal solution (whose size we characterize). Furthermore, we show that approximate consensus can be achieved geometrically fast under some minimal conditions. Our analysis provides insights into the relationship among the convergence region, distance between regular agents' values, step-size, and properties of the agents' functions for Byzantine-resilient distributed optimization.