  Inertial algorithms for minimizing nonsmooth and nonconvex functions as the
inertial proximal alternating linearized minimization algorithm (iPALM) have
demonstrated their superiority with respect to computation time over their non
inertial variants. In many problems in imaging and machine learning, the
objective functions have a special form involving huge data which encourage the
application of stochastic algorithms. While algorithms based on stochastic
gradient descent are still used in the majority of applications, recently also
stochastic algorithms for minimizing nonsmooth and nonconvex functions were
proposed. In this paper, we derive an inertial variant of a stochastic PALM
algorithm with variance-reduced gradient estimator, called iSPALM, and prove
linear convergence of the algorithm under certain assumptions. Our inertial
approach can be seen as generalization of momentum methods widely used to speed
up and stabilize optimization algorithms, in particular in machine learning, to
nonsmooth problems. Numerical experiments for learning the weights of a
so-called proximal neural network and the parameters of Student-t mixture
models show that our new algorithm outperforms both stochastic PALM and its
deterministic counterparts.
