  This paper introduces a novel algorithm, the Perturbed Proximal
Preconditioned SPIDER algorithm (3P-SPIDER), designed to solve finite sum
non-convex composite optimization. It is a stochastic Variable Metric
Forward-Backward algorithm, which allows approximate preconditioned forward
operator and uses a variable metric proximity operator as the backward
operator; it also proposes a mini-batch strategy with variance reduction to
address the finite sum setting. We show that 3P-SPIDER extends some Stochastic
preconditioned Gradient Descent-based algorithms and some Incremental
Expectation Maximization algorithms to composite optimization and to the case
the forward operator can not be computed in closed form. We also provide an
explicit control of convergence in expectation of 3P-SPIDER, and study its
complexity in order to satisfy the epsilon-approximate stationary condition.
Our results are the first to combine the composite non-convex optimization
setting, a variance reduction technique to tackle the finite sum setting by
using a minibatch strategy and, to allow deterministic or random approximations
of the preconditioned forward operator. Finally, through an application to
inference in a logistic regression model with random effects, we numerically
compare 3P-SPIDER to other stochastic forward-backward algorithms and discuss
the role of some design parameters of 3P-SPIDER.
