Distributed linearized alternating direction method of multipliers for composite convex consensus optimization

Given an undirected graph G = (N, ϵ) of agents N = {1, ⋯, N} connected with edges in ϵ, 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 {Φ_i}_i∈N, where Φ_i ≙ ξ_i + f_i belongs to agent-i. Assuming only agents connected by an edge can communicate, we propose a distributed proximal gradient algorithm (DPGA) for consensus optimization over both unweighted and weighted static (undirected) communication networks. In one iteration, each agent-i computes the prox map of ξ_i and gradient of f_i, 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 ∇f_i 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 suboptimality error and consensus violation for the DPGA and SDPGA with rates O(1/t) and O(1/√t), respectively.