Revisiting EXTRA for Smooth Distributed Optimization
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EXTRA is a popular method for the dencentralized distributed optimization and has broad applications. This paper revisits the EXTRA. Firstly, we give a sharp complexity analysis for EXTRA with the improved O((L/μ+1/1-σ_2(W))log1/ϵ(1-σ_2(W))) communication and computation complexities for μ-strongly convex and L-smooth problems, where σ_2(W) is the second largest singular value of the weight matrix W. When the strong convexity is absent, we prove the O((L/ϵ+1/1-σ_2(W))log1/1-σ_2(W)) complexities. Then, we use the Catalyst framework to accelerate EXTRA and obtain the O(√(L/μ(1-σ_2(W)))log L/μ(1-σ_2(W))log1/ϵ) communication and computation complexities for strongly convex and smooth problems and the O(√(L/ϵ(1-σ_2(W)))log1/ϵ(1-σ_2(W))) complexities for non-strongly convex ones. Our communication complexities of the accelerated EXTRA are only worse by the factors of (logL/μ(1-σ_2(W))) and (log1/ϵ(1-σ_2(W))) from the lower complexity bounds for strongly convex and non-strongly convex problems, respectively.
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