On the Performance of Gradient Tracking with Local Updates

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 and Gradient Tracking (GT) involve communicating every iteration. However, communication is expensive, resource intensive, and slow. This work analyzes a locally updated GT method (LUG T), 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 decentralized (symmetric) 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 results reveal that local updates may lead to lower communication costs in specific regimes (e.g., well-connected graphs).