This work studies a class of non-smooth decentralized multi-agent optimization problems where the agents aim at minimizing a sum of local strongly-convex smooth components plus a common non-smooth term. We propose a general algorithmic framework that captures many existing state-of-the-art algorithms including the adapt-then-combine gradient-tracking methods for smooth costs. We then prove linear convergence of the proposed method in the presence of the non-smooth term. Moreover, for the more general class of problems with agent specific non-smooth terms, we show that linear convergence cannot be achieved (in the worst case) for the class of algorithms that uses the gradients and the proximal mappings of the smooth and non-smooth parts, respectively. We further provide a numerical counterexample that shows some state-of-the-art algorithms fail to converge linearly for strongly-convex objectives and different local non-smooth terms.

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This work develops effective distributed strategies for the solution of constrained multi-agent stochastic optimization problems with coupled parameters across the agents. In this formulation, each agent is influenced by only a subset of the entries of a global parameter vector or model, and is subject to convex constraints that are only known locally. Problems of this type arise in several applications, most notably in disease propagation models, minimum-cost flow problems, distributed control formulations, and distributed power system monitoring. This work focuses on stochastic settings, where a stochastic risk function is associated with each agent and the objective is to seek the minimizer of the aggregate sum of all risks subject to a set of constraints. Agents are not aware of the statistical distribution of the data and, therefore, can only rely on stochastic approximations in their learning strategies. We derive an effective distributed learning strategy that is able to track drifts in the underlying parameter model. A detailed performance and stability analysis is carried out showing that the resulting coupled diffusion strategy converges at a linear rate to an O(µ)−neighborhood of the true penalized optimizer.

Distributed optimization methods with local updates have recently received a lot of attention due to their potential to reduce the communication cost of distributed methods. In these algorithms, a collection of nodes performs several local updates based on their local data and then communicates with each other to exchange estimate information. While there have been many studies on distributed local methods with centralized network connections, there has been less work on decentralized networks.In this work, we propose and investigate a locally updated decentralized method called Local Exact-Diffusion (LED). We establish the convergence of LED in both convex and nonconvex settings for the stochastic online setting. Our convergence rate bounds improves over the bounds of existing decentralized methods. When we specialize the network to the centralized case, we recover the state-of-the-art bound for centralized methods. We also link LED to several other distributed methods that have been studied independently, including Scaffnew, FedGate, and VRL-SGD. We also numerically investigate the benefits of local updates for decentralized networks and demonstrate the effectiveness of the proposed method.