Provably Accelerated Decentralized Gradient Method Over Unbalanced Directed Graphs

In this work, we consider the decentralized optimization problem in which a network of $n$ agents, each possessing a smooth and convex objective function, wish to collaboratively minimize the average of all the objective functions through peer-to-peer communication in a directed graph. To solve the problem, we propose two accelerated Push-DIGing methods termed APD and APD-SC for minimizing non-strongly convex objective functions and strongly convex ones, respectively. We show that APD and APD-SC respectively converge at the rates $Oleft(frac{1}{k^2}right)$ and $Oleft(left(1 - Csqrt{frac{mu}{L}}right)^kright)$ up to constant factors depending only on the mixing matrix. To the best of our knowledge, APD and APD-SC are the first decentralized methods to achieve provable acceleration over unbalanced directed graphs. Numerical experiments demonstrate the effectiveness of both methods.