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This note studies the distributed non-convex optimization problem withnon-smooth regularization, which has wide applications in decentralizedlearning, estimation and control. The objective function is the sum ofdifferent local objective functions, which consist of differentiable (possiblynon-convex) cost functions and non-smooth convex functions. This paper presentsa distributed proximal gradient algorithm for the non-smooth non-convexoptimization problem over time-varying multi-agent networks. Each agent updateslocal variable estimate by the multi-step consensus operator and the proximaloperator. We prove that the generated local variables achieve consensus andconverge to the set of critical points with convergence rate $O(1/T)$. Finally,we verify the efficacy of proposed algorithm by numerical simulations.

Distributed proximal gradient algorithm for non-smooth non-convex optimization over time-varying networks