On the Convergence of Decentralized Gradient Descent

Consider the consensus problem of minimizing $f(x)=\sum_{i=1}^n f_i(x)$, where $x\in{\mathbb{R}}^p$ and each $f_i$ is only known to the individual agent $i$ in a connected network of $n$ agents. To solve this problem and obtain the solution, all the agents collaborate with their neighbors through information exchange. This type of decentralized computation does not need a fusion center, offers better network load balance, and improves data privacy. This paper studies the decentralized gradient descent method [A. Nedic and A. Ozdaglar, IEEE Trans. Automat. Control, 54 (2009), pp. 48--61], in which each agent $i$ updates its local variable $x_{(i)}\in{\mathbb{R}}^n$ by combining the average of its neighbors'…