Provably Accelerated Decentralized Gradient Method Over Unbalanced Directed Graphs
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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 O(1/k^2) and O((1 - C√(μ/L))^k) 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.
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