Convergence results of a nested decentralized gradient method for non-strongly convex problems

TL;DR

This paper analyzes the convergence of the NEAR-DGD$^+$ method for distributed optimization problems without assuming strong convexity, extending previous results to broader classes of convex functions.

Contribution

It establishes convergence properties of NEAR-DGD$^+$ for convex and quasi-strongly convex functions, beyond the strongly convex case.

Findings

Convergence results for convex functions without strong convexity.

Convergence for composite functions with a strongly convex component.

Numerical validation of theoretical convergence results.

Abstract

We are concerned with the convergence of NEAR-DGD$^+$ (Nested Exact Alternating Recursion Distributed Gradient Descent) method introduced to solve the distributed optimization problems. Under the assumption of the strong convexity of local objective functions and the Lipschitz continuity of their gradients, the linear convergence is established in \cite{BBKW - Near DGD}. In this paper, we investigate the convergence property of NEAR-DGD$^+$ in the absence of strong convexity. More precisely, we establish the convergence results in the following two cases: (1) When only the convexity is assumed on the objective function. (2) When the objective function is represented as a composite function of a strongly convex function and a rank deficient matrix, which falls into the class of convex and quasi-strongly convex functions. Numerical results are provided to support the convergence results.