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Decentralized optimization is a powerful paradigm that finds applications inengineering and learning design. This work studies decentralized compositeoptimization problems with non-smooth regularization terms. Most existinggradient-based proximal decentralized methods are known to converge to theoptimal solution with sublinear rates, and it remains unclear whether thisfamily of methods can achieve global linear convergence. To tackle thisproblem, this work assumes the non-smooth regularization term is common acrossall networked agents, which is the case for many machine learning problems.Under this condition, we design a proximal gradient decentralized algorithmwhose fixed point coincides with the desired minimizer. We then provide aconcise proof that establishes its linear convergence. In the absence of thenon-smooth term, our analysis technique covers the well known EXTRA algorithmand provides useful bounds on the convergence rate and step-size.

A Linearly Convergent Proximal Gradient Algorithm for Decentralized Optimization