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：本文针对联邦学习场景中数据分布在大量设备上的定位问题，该问题本质上非凸且非光滑。由于联邦环境的去中心化特性，分布式学习对于可扩展性和适应性至关重要。此外，这些环境常受异常值数据干扰，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中采用基于$L_1$范数的鲁棒形式，专门设计用于处理这些障碍。我们的方法直接处理问题的原始形式，无需进行迭代简化或近似，从而提升了计算效率和估计精度。我们证明该方法收敛到稳定点，彰显其有效性和可靠性。通过数值模拟，我们验证了该方法在富含异常值的环境中性能优越，超越了现有最先进的定位方法。