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Given an undirected graph $\mathcal{G}=(\mathcal{N},\mathcal{E})$ of agents$\mathcal{N}=\{1,\ldots,N\}$ connected with edges in $\mathcal{E}$, we studyhow to compute an optimal decision on which there is consensus among agents andthat minimizes the sum of agent-specific private convex composite functions$\{\Phi_i\}_{i\in\mathcal{N}}$ while respecting privacy requirements, where$\Phi_i\triangleq \xi_i + f_i$ belongs to agent-$i$. Assuming only agentsconnected by an edge can communicate, we propose a distributed proximalgradient method DPGA for consensus optimization over both unweighted andweighted static (undirected) communication networks. In one iteration, eachagent-$i$ computes the prox map of $\xi_i$ and gradient of $f_i$, and this isfollowed by local communication with neighboring agents. We also study itsstochastic gradient variant, SDPGA, which can only access to noisy estimates of$\nabla f_i$ at each agent-$i$. This computational model abstracts a number ofapplications in distributed sensing, machine learning and statisticalinference. We show ergodic convergence in both sub-optimality error andconsensus violation for DPGA and SDPGA with rates $\mathcal{O}(1/t)$ and$\mathcal{O}(1/\sqrt{t})$, respectively.

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