Decentralized optimization over time-varying graphs has been increasingly common in modern machine learning with massive data stored on millions of mobile devices, such as in federated learning. This paper revisits and extends the widely used accelerated gradient tracking. We prove the $\cal O(\frac{\gamma^2}{(1-\sigma_{\gamma})^2}\sqrt{\frac{L}{\epsilon}})$ and $\cal O((\frac{\gamma}{1-\sigma_{\gamma}})^{1.5}\sqrt{\frac{L}{\mu}}\log\frac{1}{\epsilon})$ complexities for the practical single loop accelerated gradient tracking over time-varying graphs when the problems are nonstrongly convex and strongly convex, respectively, where $\gamma$ and $\sigma_{\gamma}$ are two common constants charactering the network connectivity, $\epsilon$ is the desired precision, and $L$ and $\mu$ are the smoothness and strong convexity constants, respectively. Our complexities improve significantly on the ones of $\cal O(\frac{1}{\epsilon^{5/7}})$ and $\cal O((\frac{L}{\mu})^{5/7}\frac{1}{(1-\sigma)^{1.5}}\log\frac{1}{\epsilon})$ proved in the original literature only for static graph. When combining with a multiple consensus subroutine, the dependence on the network connectivity constants can be further improved. When the network is time-invariant, our complexities exactly match the lower bounds without hiding any poly-logarithmic factor for both nonstrongly convex and strongly convex problems.

翻译：在现代机器学习中,对时间变化图形的分散优化越来越常见, 大量数据储存在数百万移动设备上, 比如在联盟式学习中。 本文重新审视并扩展广泛使用的加速梯度跟踪。 当问题不是强烈的 convex和强烈的 convex时, 我们证明$O( gamma$) 和$( gmama_% 1 -\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\