A First-Order Stochastic Primal-Dual Algorithm with Correction Step

We investigate the convergence properties of a stochastic primal-dual splitting algorithmfor solving structured monotone inclusions involving the sum of a cocoercive operator and acomposite monotone operator. The proposed method is the stochastic extension to monotoneinclusions of a proximal method studied in [26, 35] for saddle point problems. It consists in aforward step determined by the stochastic evaluation of the cocoercive operator, a backwardstep in the dual variables involving the resolvent of the monotone operator, and an additionalforward step using the stochastic evaluation of the cocoercive introduced in the first step. Weprove weak almost sure convergence of the iterates by showing that the primal-dual sequencegenerated by the method is stochastic quasi Fej ́er-monotone with respect to the set of zeros of theconsidered primal and dual inclusions. Additional results on ergodicconvergence in expectationare considered for the special case of saddle point models.