Nearly Optimal Regret for Decentralized Online Convex Optimization

Abstract

We investigate decentralized online convex optimization (D-OCO), in which a set of local learners are required to minimize a sequence of global loss functions using only local computations and communications. Previous studies have established $O(n^{5/4}\rho^{-1/2}\sqrt{T})$ and ${O}(n^{3/2}\rho^{-1}\log T)$ regret bounds for convex and strongly convex functions respectively, where $n$ is the number of local learners, $\rho<1$ is the spectral gap of the communication matrix, and $T$ is the time horizon. However, there exist large gaps from the existing lower bounds, i.e., $\Omega(n\sqrt{T})$ for convex functions and $\Omega(n)$ for strongly convex functions. To fill these gaps, in this paper, we first develop novel D-OCO algorithms that can respectively reduce the regret bounds for convex and strongly convex functions to $\tilde{O}(n\rho^{-1/4}\sqrt{T})$ and $\tilde{O}(n\rho^{-1/2}\log T)$. The primary technique is to design an online accelerated gossip strategy that enjoys a faster average consensus among local learners. Furthermore, by carefully exploiting the spectral properties of a specific network topology, we enhance the lower bounds for convex and strongly convex functions to $\Omega(n\rho^{-1/4}\sqrt{T})$ and $\Omega(n\rho^{-1/2})$, respectively. These lower bounds suggest that our algorithms are nearly optimal in terms of $T$, $n$, and $\rho$.