We consider the decentralized optimization problem, where a network of $n$ agents aims to collaboratively minimize the average of their individual smooth and convex objective functions through peer-to-peer communication in a directed graph. To tackle this problem, we propose two accelerated gradient tracking methods, namely APD and APD-SC, for non-strongly convex and strongly convex objective functions, respectively. We show that APD and APD-SC converge at the rates $O\left(\frac{1}{k^2}\right)$ and $O\left(\left(1 - C\sqrt{\frac{\mu}{L}}\right)^k\right)$, respectively, up to constant factors depending only on the mixing matrix. APD and APD-SC are the first decentralized methods over unbalanced directed graphs that achieve the same provable acceleration as centralized methods. Numerical experiments demonstrate the effectiveness of both methods.

翻译：我们考虑分散优化问题：在由n个智能体组成的网络中，每个智能体通过有向图进行点对点通信，旨在协同最小化各自光滑且凸的目标函数的平均值。为解决该问题，我们针对非强凸和强凸目标函数，分别提出了两种加速梯度跟踪方法——APD和APD-SC。我们证明，APD和APD-SC的收敛速率分别为$O\left(\frac{1}{k^2}\right)$和$O\left(\left(1 - C\sqrt{\frac{\mu}{L}}\right)^k\right)$，其中常数因子仅依赖于混合矩阵。APD和APD-SC是首个在非平衡有向图上实现与集中式方法相同可证明加速的分散方法。数值实验验证了这两种方法的有效性。