The recent deployment of multi-agent systems in a wide range of scenarios has enabled the solution of learning problems in a distributed fashion. In this context, agents are tasked with collecting local data and then cooperatively train a model, without directly sharing the data. While distributed learning offers the advantage of preserving agents' privacy, it also poses several challenges in terms of designing and analyzing suitable algorithms. This work focuses specifically on the following challenges motivated by practical implementation: (i) online learning, where the local data change over time; (ii) asynchronous agent computations; (iii) unreliable and limited communications; and (iv) inexact local computations. To tackle these challenges, we introduce the Distributed Operator Theoretical (DOT) version of the Alternating Direction Method of Multipliers (ADMM), which we call the DOT-ADMM Algorithm. We prove that it converges with a linear rate for a large class of convex learning problems (e.g., linear and logistic regression problems) toward a bounded neighborhood of the optimal time-varying solution, and characterize how the neighborhood depends on~$\text{(i)--(iv)}$. We corroborate the theoretical analysis with numerical simulations comparing the DOT-ADMM Algorithm with other state-of-the-art algorithms, showing that only the proposed algorithm exhibits robustness to (i)--(iv).

翻译：近期多智能体系统在各类场景中的广泛应用，使得以分布式方式解决学习问题成为可能。在此背景下，智能体需收集本地数据并协同训练模型，同时避免直接共享数据。尽管分布式学习具有保护智能体隐私的优势，但在设计及分析相应算法方面也带来诸多挑战。本研究聚焦于实际部署中由以下因素引发的具体挑战：(i) 在线学习——本地数据随时间动态变化；(ii) 异步智能体计算；(iii) 不可靠且受限的通信；(iv) 不精确的本地计算。为应对这些挑战，我们提出交替方向乘子法（ADMM）的分布式算子理论（DOT）版本，命名为DOT-ADMM算法。我们证明，该算法针对一大类凸学习问题（如线性回归与逻辑回归问题）能以线性速率收敛至最优时变解的有界邻域，并刻画该邻域对~(i)--(iv)的依赖关系。通过数值仿真将DOT-ADMM算法与当前最优算法进行对比，理论分析得到验证：仅所提算法对(i)--(iv)具有鲁棒性。