On the Convergence of NEAR-DGD for Nonconvex Optimization with Second Order Guarantees

TL;DR

This paper extends the NEAR-DGD distributed optimization method to nonconvex problems, showing convergence to approximate second-order stationary points and balancing communication and computation.

Contribution

It generalizes NEAR-DGD's convergence analysis from strongly convex to nonconvex functions, providing theoretical guarantees and empirical evaluation.

Findings

Convergence to minimizers of a Lyapunov function under mild assumptions

The gap to second-order stationary solutions can be made arbitrarily small

Numerical experiments confirm empirical performance in nonconvex settings

Abstract

We consider the setting where the nodes of an undirected, connected network collaborate to solve a shared objective modeled as the sum of smooth functions. We assume that each summand is privately known by a unique node. NEAR-DGD is a distributed first order method which permits adjusting the amount of communication between nodes relative to the amount of computation performed locally in order to balance convergence accuracy and total application cost. In this work, we generalize the convergence properties of a variant of NEAR-DGD from the strongly convex to the nonconvex case. Under mild assumptions, we show convergence to minimizers of a custom Lyapunov function. Moreover, we demonstrate that the gap between those minimizers and the second order stationary solutions of the original problem can become arbitrarily small depending on the choice of algorithm parameters. Finally, we accompany our theoretical analysis with a numerical experiment to evaluate the empirical performance of NEAR-DGD in the nonconvex setting.