  In this paper, we propose a novel Dual Inexact Splitting Algorithm (DISA) for
distributed convex composite optimization problems, where the local loss
function consists of a smooth term and a possibly nonsmooth term composed with
a linear mapping. DISA, for the first time, eliminates the dependence of the
convergent step-size range on the Euclidean norm of the linear mapping, while
inheriting the advantages of the classic Primal-Dual Proximal Splitting
Algorithm (PD-PSA): simple structure and easy implementation. This indicates
that DISA can be executed without prior knowledge of the norm, and tiny
step-sizes can be avoided when the norm is large. Additionally, we prove
sublinear and linear convergence rates of DISA under general convexity and
metric subregularity, respectively. Moreover, we provide a variant of DISA with
approximate proximal mapping and prove its global convergence and sublinear
convergence rate. Numerical experiments corroborate our theoretical analyses
and demonstrate a significant acceleration of DISA compared to existing
PD-PSAs.
