Accelerated Gradient Tracking over Time-varying Graphs for Decentralized Optimization

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.