Stochastic forward-backward and primal-dual approximation algorithms with application to online image restoration

Patrick L Combettes (1) , Jean-Christophe Pesquet (2, 3)

1 LJLL - Laboratoire Jacques-Louis Lions
2 GALEN - Organ Modeling through Extraction, Representation and Understanding of Medical Image Content
3 LIGM - Laboratoire d'Informatique Gaspard-Monge

Stochastic approximation techniques have been used in various contexts in data science. We propose a stochastic version of the forward-backward algorithm for minimizing the sum of two convex functions, one of which is not necessarily smooth. Our framework can handle stochastic approximations of the gradient of the smooth function and allows for stochastic errors in the evaluation of the proximity operator of the nonsmooth function. The almost sure convergence of the iterates generated by the algorithm to a minimizer is established under relatively mild assumptions. We also propose a stochastic version of a popular primal-dual proximal splitting algorithm, establish its convergence, and apply it to an online image restoration problem.

Mots clés: Index Terms— convex optimization; nonsmooth optimization; primal-dual algorithm; stochastic algorithm; parallel algorithm; proximity operator; recovery; image restoration

Domaines: Traitement du signal et de l'image [eess.SP]

Patrick L Combettes, Jean-Christophe Pesquet. Stochastic forward-backward and primal-dual approximation algorithms with application to online image restoration. European Signal and Image Processing Conference (EUSIPCO 2016), Aug 2016, Budapest, Hungary. pp.1813 - 1817, ⟨10.1109/EUSIPCO.2016.7760561⟩. ⟨hal-01422154⟩