Can Primal Methods Outperform Primal-Dual Methods in Decentralized Dynamic Optimization?

In this paper, we consider the decentralized dynamic optimization problem defined over a multi-agent network. Each agent possesses a time-varying local objective function, and all agents aim to collaboratively track the drifting global optimal solution that minimizes the summation of all local objective functions. The decentralized dynamic optimization problem can be solved by primal or primal-dual methods, and when the problem degenerates to be static, it has been proved in literature that primal-dual methods are superior to primal ones. This motivates us to ask: are primal-dual methods necessarily better than primal ones in decentralized dynamic optimization? To answer this question, we investigate, and compare convergence properties of the primal method, diffusion, and the primal-dual approach, decentralized gradient tracking (DGT). Theoretical analysis reveals that diffusion can outperform DGT in certain dynamic settings. We find that DGT, and diffusion are significantly affected by the drifts, and the magnitudes of optimal gradients, respectively. In addition, we show that DGT is more sensitive to the network topology, and a badly-connected network can greatly deteriorate its convergence performance. These conclusions provide guidelines on how to choose proper dynamic algorithms in various application scenarios. Numerical experiments are constructed to validate the theoretical analysis.