  We study the decentralized optimization problem where a network of $n$ agents
seeks to minimize the average of a set of heterogeneous non-convex cost
functions distributedly. State-of-the-art decentralized algorithms like Exact
Diffusion~(ED) and Gradient Tracking~(GT) involve communicating every
iteration. However, communication is expensive, resource intensive, and slow.
In this work, we analyze a locally updated GT method (LU-GT), where agents
perform local recursions before interacting with their neighbors. While local
updates have been shown to reduce communication overhead in practice, their
theoretical influence has not been fully characterized. We show LU-GT has the
same communication complexity as the Federated Learning setting but allows
arbitrary network topologies. In addition, we prove that the number of local
updates does not degrade the quality of the solution achieved by LU-GT.
Numerical examples reveal that local updates can lower communication costs in
certain regimes (e.g., well-connected graphs).
