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This paper develops a fully distributed differentially-private learningalgorithm to solve nonsmooth optimization problems. We distribute theAlternating Direction Method of Multipliers (ADMM) to comply with thedistributed setting and employ an approximation of the augmented Lagrangian tohandle nonsmooth objective functions. Furthermore, we ensure zero-concentrateddifferential privacy (zCDP) by perturbing the outcome of the computation ateach agent with a variance-decreasing Gaussian noise. This privacy-preservingmethod allows for better accuracy than the conventional $(\epsilon, \delta)$-DPand stronger guarantees than the more recent R\'enyi-DP. The developed fullydistributed algorithm has a competitive privacy accuracy trade-off and handlesnonsmooth and non-necessarily strongly convex problems. We provide completetheoretical proof for the privacy guarantees and the convergence of thealgorithm to the exact solution. We also prove under additional assumptionsthat the algorithm converges in linear time. Finally, we observe in simulationsthat the developed algorithm outperforms all of the existing methods.

Zero-Concentrated Private Distributed Learning for Nonsmooth Objective Functions