A Fully Stochastic Primal-Dual Algorithm

A new stochastic primal-dual algorithm for solving a composite optimization problem is proposed. It is assumed that all the functions/matrix used to define the optimization problem are given as statistical expectations. These expectations are unknown but revealed across time through i.i.d realizations. This covers the case of convex optimization under stochastic linear constraints. The proposed algorithm is proven to converge to a saddle point of the Lagrangian function. In the framework of the monotone operator theory, the convergence proof relies on recent results on the stochastic Forward Backward algorithm involving random monotone operators.