  We study a version of the proximal gradient algorithm for which the gradient
is intractable and is approximated by Monte Carlo methods (and in particular
Markov Chain Monte Carlo). We derive conditions on the step size and the Monte
Carlo batch size under which convergence is guaranteed: both increasing batch
size and constant batch size are considered. We also derive non-asymptotic
bounds for an averaged version. Our results cover both the cases of biased and
unbiased Monte Carlo approximation. To support our findings, we discuss the
inference of a sparse generalized linear model with random effect and the
problem of learning the edge structure and parameters of sparse undirected
graphical models.
