Linear Time Average Consensus and Distributed Optimization on Fixed Graphs

We describe a protocol for the average consensus problem on any fixed undirected graph whose convergence time scales linearly in the total number nodes $n$. The protocol relies only on nearest-neighbor interactions but requires all the nodes to know the same upper bound $U$ on the total number of nodes which is correct within a constant multiplicative factor. As an application, we develop a distributed protocol for minimizing an average of (possibly nondifferentiable) convex functions $(1/n) \sum_{i=1}^n f_i(\theta)$ in the setting where only node $i$ in an undirected, connected graph knows the function $f_i(\theta)$. Under the same assumption about all nodes knowing $U$, and additionally assuming that the subgradients of each $f_i(\theta)$ have absolute values bounded by some constant $L$ known to the nodes, we show that after $T$ iterations our protocol has error which is $O(L \sqrt{n/T})$. As a consequence, the time to solve this distributed optimization problem to any fixed accuracy is also linear in ...