  We study the block-coordinate forward-backward algorithm in which the blocks
are updated in a random and possibly parallel manner, according to arbitrary
probabilities. The algorithm allows different stepsizes along the
block-coordinates to fully exploit the smoothness properties of the objective
function. In the convex case and in an infinite dimensional setting, we
establish almost sure weak convergence of the iterates and the asymptotic rate
o(1/n) for the mean of the function values. We derive linear rates under strong
convexity and error bound conditions. Our analysis is based on an abstract
convergence principle for stochastic descent algorithms which allows to extend
and simplify existing results.
