  In decentralized learning, a network of nodes cooperate to minimize an
overall objective function that is usually the finite-sum of their local
objectives, and incorporates a non-smooth regularization term for the better
generalization ability. Decentralized stochastic proximal gradient (DSPG)
method is commonly used to train this type of learning models, while the
convergence rate is retarded by the variance of stochastic gradients. In this
paper, we propose a novel algorithm, namely DPSVRG, to accelerate the
decentralized training by leveraging the variance reduction technique. The
basic idea is to introduce an estimator in each node, which tracks the local
full gradient periodically, to correct the stochastic gradient at each
iteration. By transforming our decentralized algorithm into a centralized
inexact proximal gradient algorithm with variance reduction, and controlling
the bounds of error sequences, we prove that DPSVRG converges at the rate of
$O(1/T)$ for general convex objectives plus a non-smooth term with $T$ as the
number of iterations, while DSPG converges at the rate $O(\frac{1}{\sqrt{T}})$.
Our experiments on different applications, network topologies and learning
models demonstrate that DPSVRG converges much faster than DSPG, and the loss
function of DPSVRG decreases smoothly along with the training epochs.
