Distributed Online Convex Optimization with Efficient Communication: Improved Algorithm and Lower bounds

We investigate distributed online convex optimization with compressed communication, where $n$ learners connected by a network collaboratively minimize a sequence of global loss functions using only local information and compressed data from neighbors. Prior work has established regret bounds of $O(\max\{\omega^{-2}\rho^{-4}n^{1/2},\omega^{-4}\rho^{-8}\}n\sqrt{T})$ and $O(\max\{\omega^{-2}\rho^{-4}n^{1/2},\omega^{-4}\rho^{-8}\}n\ln{T})$ for convex and strongly convex functions, respectively, where $\omega\in(0,1]$ is the compression quality factor ($\omega=1$ means no compression) and $\rho<1$ is the spectral gap of the communication matrix. However, these regret bounds suffer from a \emph{quadratic} or even \emph{quartic} dependence on $\omega^{-1}$. Moreover, the \emph{super-linear} dependence on $n$ is also undesirable. To overcome these limitations, we propose a novel algorithm that achieves improved regret bounds of $\tilde{O}(\omega^{-1/2}\rho^{-1}n\sqrt{T})$ and $\tilde{O}(\omega^{-1}\rho^{-2}n\ln{T})$ for convex and strongly convex functions, respectively. The primary idea is to design a \emph{two-level blocking update framework} incorporating two novel ingredients: an online gossip strategy and an error compensation scheme, which collaborate to \emph{achieve a better consensus} among learners. Furthermore, we establish the first lower bounds for this problem, justifying the optimality of our results with respect to both $\omega$ and $T$. Additionally, we consider the bandit feedback scenario, and extend our method with the classic gradient estimators to enhance existing regret bounds.