  EXTRA is a popular method for dencentralized distributed optimization and has
broad applications. This paper revisits EXTRA. First, we give a sharp
complexity analysis for EXTRA with the improved
$O\left(\left(\frac{L}{\mu}+\frac{1}{1-\sigma_2(W)}\right)\log\frac{1}{\epsilon(1-\sigma_2(W))}\right)$
communication and computation complexities for $\mu$-strongly convex and
$L$-smooth problems, where $\sigma_2(W)$ is the second largest singular value
of the weight matrix $W$. When the strong convexity is absent, we prove the
$O\left(\left(\frac{L}{\epsilon}+\frac{1}{1-\sigma_2(W)}\right)\log\frac{1}{1-\sigma_2(W)}\right)$
complexities. Then, we use the Catalyst framework to accelerate EXTRA and
obtain the $O\left(\sqrt{\frac{L}{\mu(1-\sigma_2(W))}}\log\frac{
L}{\mu(1-\sigma_2(W))}\log\frac{1}{\epsilon}\right)$ communication and
computation complexities for strongly convex and smooth problems and the
$O\left(\sqrt{\frac{L}{\epsilon(1-\sigma_2(W))}}\log\frac{1}{\epsilon(1-\sigma_2(W))}\right)$
complexities for non-strongly convex ones. Our communication complexities of
the accelerated EXTRA are only worse by the factors of
$\left(\log\frac{L}{\mu(1-\sigma_2(W))}\right)$ and
$\left(\log\frac{1}{\epsilon(1-\sigma_2(W))}\right)$ from the lower complexity
bounds for strongly convex and non-strongly convex problems, respectively.
