LAUSR

In this article, we consider a multi-node sharing problem, where each node possesses a local smooth function that is further considered as the average of several constituent functions, and the network aims to minimize a finite-sum of all local functions plus a coupling function (possibly non-smooth). Decentralized optimization to solve this problem has been a significant focus within engineering research due to its advantages in scalability, robustness, and flexibility. To this aim, an equivalent saddle-point problem of this problem is first formulated, which is amenable to decentralized solutions. Then, a novel decentralized stochastic algorithm, named VR-DPPD, is proposed, which combines the variance-reduction technique of SAGA with the decentralized proximal primal-dual method. We provide a convergence analysis and show that VR-DPPD converges linearly to the exact optimal solution in expectation if smooth local functions are strongly convex. Our work makes progress towards resolving a general composite optimization problem with a convex (possibly non-smooth) coupling function, giving a novel linear convergent algorithm for achieving low computation cost. Numerical examples are presented to demonstrate the viability and performance of VR-DPPD.