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 $O(\max\{ω^{-2}ρ^{-4}n^{1/2},ω^{-4}ρ^{-8}\}n\sqrt{T})$ and $O(\max\{ω^{-2}ρ^{-4}n^{1/2},ω^{-4}ρ^{-8}\}n\ln{T})$ for convex and strongly convex functions, respectively, where $ω\in(0,1]$ is the compression quality factor ($ω=1$ means no compression) and $ρ<1$ is the spectral gap of the communication matrix. However, these regret bounds suffer from a \emph{quadratic} or even \emph{quartic} dependence on $ω^{-1}$. 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 $\tilde{O}(ω^{-1/2}ρ^{-1}n\sqrt{T})$ and $\tilde{O}(ω^{-1}ρ^{-2}n\ln{T})$ 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 $ω$ and $T$. Additionally, we consider the bandit feedback scenario, and extend our method with the classic gradient estimators to enhance existing regret bounds.

翻译：本文研究压缩通信下的分布式在线凸优化问题，其中$n$个学习者通过网络连接，仅利用本地信息和来自邻居的压缩数据协同最小化一系列全局损失函数。已有工作针对凸函数和强凸函数分别建立了$O(\max\{ω^{-2}ρ^{-4}n^{1/2},ω^{-4}ρ^{-8}\}n\sqrt{T})$与$O(\max\{ω^{-2}ρ^{-4}n^{1/2},ω^{-4}ρ^{-8}\}n\ln{T})$的遗憾界，其中$ω\in(0,1]$为压缩质量因子（$ω=1$表示无压缩），$ρ<1$为通信矩阵的谱间隙。然而，这些遗憾界对$ω^{-1}$存在\emph{二次}甚至\emph{四次}依赖，且对$n$的\emph{超线性}依赖亦不理想。为克服这些局限，我们提出一种新算法，对凸函数和强凸函数分别实现$\tilde{O}(ω^{-1/2}ρ^{-1}n\sqrt{T})$与$\tilde{O}(ω^{-1}ρ^{-2}n\ln{T})$的改进遗憾界。其核心思想是设计一种\emph{双层分块更新框架}，融合在线随机通信策略与误差补偿机制两项创新要素，协同实现学习者间\emph{更优的共识}。此外，我们首次建立了该问题的下界，验证了所得结果在$ω$与$T$维度上的最优性。进一步，针对赌博机反馈场景，我们结合经典梯度估计器扩展了所提方法，改进了现有遗憾界。