Linear Time Average Consensus on Fixed Graphs and Implications for Decentralized Optimization and Multi-Agent Control

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 is completely distributed, with the exception of requiring all nodes to know the same upper bound $U$ on the total number of nodes which is correct within a constant multiplicative factor. We next discuss applications of this protocol to problems in multi-agent control connected to the consensus problem. In particular, we describe protocols for formation maintenance and leader-following with convergence times which also scale linearly with the number of nodes. Finally, 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 upper bounded by some constant $L$ known to the nodes, we show that after $T$ iterations our protocol has error which is $O(L \sqrt{n/T})$.