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 $\mu.$ 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 $\mu$ 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.

翻译：我们提出了一种基于观测矩阵的框架，用于应对对抗者的完全异步在线联邦学习。在本工作中，我们证明了该框架在估计随机向量均值方面的有效性。主要结果是，所提算法几乎必然收敛到目标均值 $\mu$。这使得该方法成为首个在对抗存在下具有几乎必然收敛保证的异步联邦学习算法。我们通过基于微分包含的双时间尺度分析推导出这一收敛性。证明中的另外两个亮点包括：(a) 使用新型李雅普诺夫函数证明 $\mu$ 是算法极限动力学的唯一全局吸引子；(b) 结合鞅与停时理论证明算法迭代量几乎必然有界。