  The nonconvex and nonsmooth finite-sum optimization problem with linear
constraint has attracted much attention in the fields of artificial
intelligence, computer, and mathematics, due to its wide applications in
machine learning and the lack of efficient algorithms with convincing
convergence theories. A popular approach to solve it is the stochastic
Alternating Direction Method of Multipliers (ADMM), but most stochastic
ADMM-type methods focus on convex models. In addition, the variance reduction
(VR) and acceleration techniques are useful tools in the development of
stochastic methods due to their simplicity and practicability in providing
acceleration characteristics of various machine learning models. However, it
remains unclear whether accelerated SVRG-ADMM algorithm (ASVRG-ADMM), which
extends SVRG-ADMM by incorporating momentum techniques, exhibits a comparable
acceleration characteristic or convergence rate in the nonconvex setting. To
fill this gap, we consider a general nonconvex nonsmooth optimization problem
and study the convergence of ASVRG-ADMM. By utilizing a well-defined potential
energy function, we establish its sublinear convergence rate $O(1/T)$, where
$T$ denotes the iteration number. Furthermore, under the additional
Kurdyka-Lojasiewicz (KL) property which is less stringent than the frequently
used conditions for showcasing linear convergence rates, such as strong
convexity, we show that the ASVRG-ADMM sequence has a finite length and
converges to a stationary solution with a linear convergence rate. Several
experiments on solving the graph-guided fused lasso problem and regularized
logistic regression problem validate that the proposed ASVRG-ADMM performs
better than the state-of-the-art methods.
