  Decentralized optimization is a powerful paradigm that finds applications in
engineering and learning design. This work studies decentralized composite
optimization problems with non-smooth regularization terms. Most existing
gradient-based proximal decentralized methods are known to converge to the
optimal solution with sublinear rates, and it remains unclear whether this
family of methods can achieve global linear convergence. To tackle this
problem, this work assumes the non-smooth regularization term is common across
all networked agents, which is the case for many machine learning problems.
Under this condition, we design a proximal gradient decentralized algorithm
whose fixed point coincides with the desired minimizer. We then provide a
concise proof that establishes its linear convergence. In the absence of the
non-smooth term, our analysis technique covers the well known EXTRA algorithm
and provides useful bounds on the convergence rate and step-size.
