This paper addresses the problem of localization, which is inherently non-convex and non-smooth in a federated setting where the data is distributed across a multitude of devices. Due to the decentralized nature of federated environments, distributed learning becomes essential for scalability and adaptability. Moreover, these environments are often plagued by outlier data, which presents substantial challenges to conventional methods, particularly in maintaining estimation accuracy and ensuring algorithm convergence. To mitigate these challenges, we propose a method that adopts an $L_1$-norm robust formulation within a distributed sub-gradient framework, explicitly designed to handle these obstacles. Our approach addresses the problem in its original form, without resorting to iterative simplifications or approximations, resulting in enhanced computational efficiency and improved estimation accuracy. We demonstrate that our method converges to a stationary point, highlighting its effectiveness and reliability. Through numerical simulations, we confirm the superior performance of our approach, notably in outlier-rich environments, which surpasses existing state-of-the-art localization methods.

翻译：本文研究了定位问题，该问题在数据分布于众多设备的联邦场景中本质上是非凸且非光滑的。由于联邦环境的去中心化特性，分布式学习对可扩展性和适应性至关重要。此外，这些环境常受异常数据困扰，这对传统方法构成重大挑战，尤其在保持估计精度和确保算法收敛方面。为缓解这些挑战，我们提出了一种方法，在分布式次梯度框架内采用L1范数鲁棒公式，专门设计用于应对这些障碍。我们的方法以其原始形式处理问题，无需借助迭代简化或近似，从而提高了计算效率和估计精度。我们证明了该方法收敛到一个平稳点，凸显了其有效性和可靠性。通过数值模拟，我们确认了该方法在异常值丰富的环境中具有优于现有最先进定位方法的性能。