# Online Learning with Adversaries: A Differential-Inclusion Analysis

arXiv: https://arxiv.org/abs/2304.01525 · HTML: https://mlsanity.com/paper/2304.01525/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 28 references that are in the map.
- First posted on arXiv 4 Apr 2023.

## Builds on (most cited first, 12 of 12)

- [Communication-Efficient Learning of Deep Networks from Decentralized Data](https://mlsanity.com/paper/1602.05629/) (2016)
- [Byzantine-Robust Distributed Learning: Towards Optimal Statistical Rates](https://mlsanity.com/paper/1803.01498/) (2018)
- [Federated Reinforcement Learning: Techniques, Applications, and Open Challenges](https://mlsanity.com/paper/2108.11887/) (2021)
- [Federated Variance-Reduced Stochastic Gradient Descent with Robustness to Byzantine Attacks](https://mlsanity.com/paper/1912.12716/) (2019)
- [Zeno++: Robust Fully Asynchronous SGD](https://mlsanity.com/paper/1903.07020/) (2019)
- [Stochastic-Sign SGD for Federated Learning with Theoretical Guarantees](https://mlsanity.com/paper/2002.10940/) (2020)
- [AFLGuard: Byzantine-robust Asynchronous Federated Learning](https://mlsanity.com/paper/2212.06325/) (2022)
- [Byzantine-Resilient High-Dimensional Federated Learning](https://mlsanity.com/paper/2006.13041/) (2020)
- [Data Encoding for Byzantine-Resilient Distributed Optimization](https://mlsanity.com/paper/1907.02664/) (2019)
- [Stochastic Recursive Inclusions in two timescales with non-additive iterate dependent Markov noise](https://mlsanity.com/paper/1611.05961/) (2016)
- [Secure State Estimation against Sensor Attacks in the Presence of Noise](https://mlsanity.com/paper/1510.02462/) (2015)
- [A Law of Iterated Logarithm for Multi-Agent Reinforcement Learning](https://mlsanity.com/paper/2110.15092/) (2021)

## Built on by (most cited first, 3 of 3)

- [Byzantine-Robust Distributed Online Learning: Taming Adversarial Participants in An Adversarial Environment](https://mlsanity.com/paper/2307.07980/) (2023)
- [Tight Convergence Rates for Online Distributed Linear Estimation with Adversarial Measurements](https://mlsanity.com/paper/2604.06282/) (2026)
- [Adversary-Robust Learning from Fully Asynchronous Directional Derivative Estimates](https://mlsanity.com/paper/2605.09337/) (2026)

## Abstract (arXiv)

We introduce an observation-matrix-based framework for fully asynchronous online Federated Learning (FL) with adversaries. In this work, we demonstrate its effectiveness in estimating the mean of a random vector. Our main result is that the proposed algorithm almost surely converges to the desired mean μ. This makes ours the first asynchronous FL method to have an a.s. convergence guarantee in the presence of adversaries. We derive this convergence using a novel differential-inclusion-based two-timescale analysis. Two other highlights of our proof include (a) the use of a novel Lyapunov function to show that μ is the unique global attractor for our algorithm's limiting dynamics, and (b) the use of martingale and stopping-time theory to show that our algorithm's iterates are almost surely bounded.