A Unifying Primal-Dual Proximal Framework for Distributed Nonconvex Optimization

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.