A stochastic inertial forward–backward splitting algorithm for multivariate monotone inclusions

Abstract

We propose an inertial forward–backward splitting algorithm to compute a zero of a sum of two monotone operators allowing for stochastic errors in the computation of the operators. More precisely, we establish almost sure convergence in real Hilbert spaces of the sequence of iterates to an optimal solution. Then, based on this analysis, we introduce two new classes of stochastic inertial primal–dual splitting methods for solving structured systems of composite monotone inclusions and prove their convergence. Our results extend to the stochastic and inertial setting various types of structured monotone inclusion problems and corresponding algorithmic solutions. Application to minimization problems is discussed.

Keywords:

• Monotone inclusion
• monotone operator
• operator splitting
• cocoercive operator
• forward–backward algorithm
• composite operator
• duality
• primal–dual algorithm

AMS Subject Classifications:

• 47H05
• 49M29
• 49M27
• 90C25

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This material is based upon work supported by the Center for Brains, Minds and Machines (CBMM), funded by NSF STC award [CCF-1231216]. L. Rosasco acknowledges the financial support of the Italian Ministry of Education, University and Research FIRB project [RBFR12M3AC]. S. Villa is member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). Bang Công Vu’s research work is partially funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) [grant number 102.01-2014.02].