  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.
