Improved dynamic regret of distributed online multiple Frank-Wolfe convex optimization

In this paper, we explore a distributed online convex optimization problem over a time-varying multi-agent network. The network aims to minimize a global loss function through local computation and communication with neighboring agents. To effectively handle the optimization problem which involves high-dimensional and structural constraint sets, we develop a distributed online multiple Frank-Wolfe algorithm that circumvents the expensive computational cost associated with projection operations. The dynamic regret bounds are established as \(\mathcal{O}(T^{1-\gamma}+H_{T})\) $\mathcal{O}(T^{1-\gamma }+H_T)$ with the linear oracle number \(\mathcal{O}(T^{1+\gamma})\) $\mathcal{O}(T^{1+\gamma })$ , which depends on the horizon (total iteration number) T, the function variation H_T, and the tuning parameter 0 < γ < 1. In particular, when the prior knowledge of H_T and T is available, the bound can be enhanced to \(\mathcal{O}(1+H_{T})\) $\mathcal{O}(1+H_T)$ . Moreover, we explore the significant advantages provided by the multiple iteration technique and reveal a trade-off between dynamic regret bound, computational cost, and communication cost. Finally, the performance of our algorithm is validated and compared through the distributed online ridge regression problems with two constraint sets.