On Nonconvex Decentralized Gradient Descent

Consensus optimization has received considerable attention in recent years. A number of decentralized algorithms have been proposed for {convex} consensus optimization. However, on \emph{nonconvex} consensus optimization, our understanding to the behavior of these algorithms is limited. This note first analyzes the convergence of the algorithm Decentralized Gradient Descent (DGD) applied to a consensus optimization problem with a smooth, possibly nonconvex objective function. We use a fixed step size under a proper bound and establish that the DGD iterates converge to a stationary point of a Lyapunov function, which approximates one of the original problem. The difference between each local point and their global average is subject to a bound proportional to the step size. This note then establishes similar results for the algorithm Prox-DGD, which is designed to minimize the sum of a differentiable function and a proximable function. While both functions can be nonconvex, a larger fixed step size is allowed if the proximable function is convex.