Distributed Linearized Alternating Direction Method of Multipliers for Composite Convex Consensus Optimization

Given an undirected graph G=(N,E) of agents N=1,ldots,N connected with edges in E, we study how to compute an optimal decision on which there is consensus among agents and that minimizes the sum of agent-specific private convex composite functions Phiii∈N while respecting privacy requirements, where Phiitriangleq xii + fi belongs to agent-i. Assuming only agents connected by an edge can communicate, we propose a distributed proximal gradient method DPGA for consensus optimization over both unweighted and weighted static (undirected) communication networks. In one iteration, each agent-i computes the prox map of xii and gradient of fi, and this is followed by local communication with neighboring agents. We also study its stochastic gradient variant, SDPGA, which can only access to noisy estimates of nabla fi at each agent-i. This computational model abstracts a number of applications in distributed sensing, machine learning and statistical inference. We show ergodic convergence in both sub-optimality error and consensus violation for DPGA and SDPGA with rates O(1/t) and O(1/√t), respectively.