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 $\rho$.

翻译：我们研究了去中心化在线凸优化（D-OCO），其中一组本地学习器需要仅通过本地计算和通信来最小化一系列全局损失函数。已有研究针对凸函数和强凸函数分别建立了 $O(n^{5/4}\rho^{-1/2}\sqrt{T})$ 和 ${O}(n^{3/2}\rho^{-1}\log T)$ 的遗憾界，其中 $n$ 是本地学习器数量，$\rho<1$ 是通信矩阵的谱间隙，$T$ 是时间范围。然而，现有结果与已知下界存在较大差距，即凸函数的 $\Omega(n\sqrt{T})$ 和强凸函数的 $\Omega(n)$ 下界。为弥合这些差距，本文首先提出了新型D-OCO算法，分别将凸函数和强凸函数的遗憾界降低至 $\tilde{O}(n\rho^{-1/4}\sqrt{T})$ 和 $\tilde{O}(n\rho^{-1/2}\log T)$。核心技术在于设计一种在线加速八卦策略，该策略能在本地学习器之间实现更快的平均共识。此外，通过仔细利用特定网络拓扑的谱特性，我们将凸函数和强凸函数的理论下界分别提升至 $\Omega(n\rho^{-1/4}\sqrt{T})$ 和 $\Omega(n\rho^{-1/2})$。这些下界表明我们的算法在 $T$、$n$ 和 $\rho$ 意义上近乎最优。