We study distributed composite optimization over networks: agents minimize a sum of smooth (strongly) convex functions–the agents’ sum-utility–plus a nonsmooth (extended-valued) convex one. We propose a general unified algorithmic framework for such a class of problems and provide a convergence analysis leveraging the theory of operator splitting. Distinguishing features of our scheme are: (i) When each of the agent’s functions is strongly convex, the algorithm converges at a linear rate, whose dependence on the agents’ functions and network topology is decoupled; (ii) When the objective function is convex (but not strongly convex), similar decoupling as in (i) is established for the coefficient of the proved sublinear rate. This also reveals the role of function heterogeneity on the convergence rate. (iii) The algorithm can adjust the ratio between the number of communications and computations to achieve a rate (in terms of computations) independent on the network connectivity; and (iv) A by-product of our analysis is a tuning recommendation for several existing (non-accelerated) distributed algorithms yielding provably faster (worst-case) convergence rate for the class of problems under consideration.

中文翻译：

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复合优化的分布式算法：统一框架和收敛性分析


我们研究网络上的分布式复合优化：代理最小化平滑（强）凸函数的总和（代理的效用和）加上非平滑（扩展值）凸函数。我们针对此类问题提出了一个通用的统一算法框架，并利用算子分裂理论提供了收敛分析。我们的方案的显着特征是：（i）当每个代理函数都是强凸函数时，算法以线性速率收敛，其对代理函数和网络拓扑的依赖性是解耦的； (ii) 当目标函数是凸函数（但不是强凸函数）时，对于证明的次线性率的系数建立与（i）中类似的解耦。这也揭示了函数异质性对收敛速度的作用。 (iii) 该算法可以调整通信次数和计算次数之间的比率，以实现独立于网络连接性的速率（就计算而言）； (iv) 我们分析的副产品是对几种现有（非加速）分布式算法的调整建议，对于所考虑的问题类别，可以证明更快（最坏情况）的收敛速度。