A Unifying Primal-Dual Proximal Framework for Distributed Nonconvex Optimization

TL;DR

This paper introduces a unifying primal-dual proximal framework for distributed nonconvex optimization, achieving improved convergence guarantees and communication efficiency through specialized algorithms and acceleration techniques.

Contribution

The paper develops a unifying framework that encompasses various existing methods, introduces new algorithms with provable convergence rates, and enhances communication efficiency via Chebyshev acceleration.

Findings

Both UPP-MC and UPP-SC achieve stationary solutions at sublinear rates.

Under P-{\

Abstract

We consider distributed nonconvex optimization over an undirected network, where each node privately possesses its local objective and communicates exclusively with its neighboring nodes, striving to collectively achieve a common optimal solution. To handle the nonconvexity of the objective, we linearize the augmented Lagrangian function and introduce a time-varying proximal term. This approach leads to a Unifying Primal-Dual Proximal (UPP) framework that unifies a variety of existing first-order and second-order methods. Building on this framework, we further derive two specialized realizations with different communication strategies, namely UPP-MC and UPP-SC. We prove that both UPP-MC and UPP-SC achieve stationary solutions for nonconvex smooth problems at a sublinear rate. Furthermore, under the additional Polyak-{\L}ojasiewics (P-{\L}) condition, UPP-MC is linearly convergent to the global optimum. These convergence results provide new or improved guarantees for many existing methods that can be viewed as specializations of UPP-MC or UPP-SC. To further optimize the mixing process, we incorporate Chebyshev acceleration into UPP-SC, resulting in UPP-SC-OPT, which attains an optimal communication complexity bound. Extensive experiments across diverse network topologies demonstrate that our proposed algorithms outperform state-of-the-art methods in both convergence speed and communication efficiency.