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We present a Markov-chain analysis of blockwise-stochastic algorithms forsolving partially block-separable optimization problems. Our main contributionsto the extensive literature on these methods are statements about the Markovoperators and distributions behind the iterates of stochastic algorithms, andin particular the regularity of Markov operators and rates of convergence ofthe distributions of the corresponding Markov chains. This provides a detailedcharacterization of the moments of the sequences beyond just the expectedbehavior. This also serves as a case study of how randomization restoresfavorable properties to algorithms that iterations of only partial informationdestroys. We demonstrate this on stochastic blockwise implementations of theforward-backward and Douglas-Rachford algorithms for nonconvex (and, as aspecial case, convex), nonsmooth optimization.

Convergence in Distribution of Randomized Algorithms: The Case of Partially Separable Optimization