The method of randomized Bregman projections for stochastic feasibility problems

In this work, we study the method of randomized Bregman projections for stochastic convex feasibility problems, possibly with an infinite number of sets, in Euclidean spaces. Under very general assumptions, we prove almost sure convergence of the iterates to a random almost common point of the sets. We then analyze in depth the case of affine sets showing that the iterates converge Q-linearly and providing also global and local rates of convergence. This work generalizes recent developments in randomized methods for the solution of linear systems based on orthogonal projection methods. We provided several applications: sketch & project methods for solving linear systems of equations, positive definite matrix completion problem, gossip algorithms for networks consensus, the assessment of robust stability of dynamical systems, and computational solutions for multimarginal optimal transport.