In this article, we investigate the convergence properties of a stochastic primal-dual splitting algorithm for solving structured monotone inclusions involving the sum of a cocoercive operator and a composite monotone operator. The proposed method is the stochastic extension to monotone inclusions of a proximal method studied in the literature for saddle point problems. It consists in a forward step determined by the stochastic evaluation of the cocoercive operator, a backward step in the dual variables involving the resolvent of the monotone operator, and an additional forward step using the stochastic evaluation of the cocoercive operator introduced in the first step. We prove weak almost sure convergence of the iterates by showing that the primal-dual sequence generated by the method is stochastic quasi-Fejér-monotone with respect to the set of zeros of the considered primal and dual inclusions. Additional results on ergodic convergence in expectation are considered for the special case of saddle point models.

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

Villa, Silvia;
2017

Abstract

In this article, we investigate the convergence properties of a stochastic primal-dual splitting algorithm for solving structured monotone inclusions involving the sum of a cocoercive operator and a composite monotone operator. The proposed method is the stochastic extension to monotone inclusions of a proximal method studied in the literature for saddle point problems. It consists in a forward step determined by the stochastic evaluation of the cocoercive operator, a backward step in the dual variables involving the resolvent of the monotone operator, and an additional forward step using the stochastic evaluation of the cocoercive operator introduced in the first step. We prove weak almost sure convergence of the iterates by showing that the primal-dual sequence generated by the method is stochastic quasi-Fejér-monotone with respect to the set of zeros of the considered primal and dual inclusions. Additional results on ergodic convergence in expectation are considered for the special case of saddle point models.
NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION
Cocoercive operator; composite operator; duality; maximal monotone operator; monotone inclusion; operator splitting; primal-dual algorithm; stochastic errors; Analysis; Signal Processing; Computer Science Applications1707 Computer Vision and Pattern Recognition; Control and Optimization
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11311/1036368
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