  This paper proposes a novel CTA (Combine-Then-Adapt)-based decentralized
algorithm for solving convex composite optimization problems over undirected
and connected networks. The local loss function in these problems contains both
smooth and nonsmooth terms. The proposed algorithm uses uncoordinated
network-independent constant stepsizes and only needs to approximately solve a
sequence of proximal mappings, which is advantageous for solving decentralized
composite optimization problems where the proximal mappings of the nonsmooth
loss functions may not have analytical solutions. For the general convex case,
we prove an O(1/k) convergence rate of the proposed algorithm, which can be
improved to o(1/k) if the proximal mappings are solved exactly. Furthermore,
with metric subregularity, we establish a linear convergence rate for the
proposed algorithm. Numerical experiments demonstrate the efficiency of the
algorithm.
