On the Comparison between Primal and Primal-dual Methods in Decentralized Dynamic Optimization

This paper considers 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 consensus 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 strategies are superior to primal ones. This motivates us to ask: are primal-dual strategies necessarily better than primal ones in decentralized dynamic optimization?To answer this question, this paper studies and compares the convergence properties of the primal approach, decentralized gradient descent (DGD), and the primal-dual approach, decentralized gradient tracking (DGT). Theoretical analysis reveals that primal methods can outperform primal-dual methods in some dynamic settings. We find that DGT is highly influenced by the variation of optimal gradients while DGD is greatly affected by the upper bound of optimal gradients. Moreover, we show that DGT is more sensitive to the network topology and a sparsely-connected network can significantly 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.