Improved Dynamic Regret of Distributed Online Multiple Frank-Wolfe Convex Optimization

Abstract

In this paper, we consider a distributed online convex optimization problem over a time-varying multi-agent network. The goal of this network is to minimize a global loss function through local computation and communication with neighbors. To effectively handle the optimization problem with a high-dimensional and structural constraint set, we develop a distributed online multiple Frank-Wolfe algorithm to avoid the expensive computational cost of projection operation. The dynamic regret bounds are established as $\mathcal{O}(T^{1-\gamma}+H_T)$ with the linear oracle number $\mathcal{O} (T^{1+\gamma})$, which depends on the horizon (total iteration number) $T$, the function variation $H_T$, and the tuning parameter $0<\gamma<1$. In particular, when the prior knowledge of $H_T$ and $T$ is available, the bound can be enhanced to $\mathcal{O} (1+H_T)$. Moreover, we illustrate the significant advantages of 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 verified and compared through the distributed online ridge regression problems with two constraint sets.