Stochastic Approximations and Perturbations in Forward-Backward Splitting for Monotone Operators

By Patrick L. Combettes, Jean-Christophe Pesquet. 2015

62 citations recorded by the source corpus.

We investigate the asymptotic behavior of a stochastic version of the forward-backward splitting algorithm for finding a zero of the sum of a maximally monotone set-valued operator and a cocoercive operator in Hilbert spaces. Our general setting features stochastic approximations of the cocoercive operator and stochastic perturbations in the evaluation of the resolvents of the set-valued operator. In addition, relaxations and not necessarily vanishing proximal parameters are allowed. Weak and strong almost sure convergence properties of the iterates is established under mild conditions on the underlying stochastic processes. Leveraging these results, we also establish the almost sure convergence of the iterates of a stochastic variant of a primal-dual proximal splitting method for composite minimization problems.

Subfields: Computational Mechanics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d26fb5134e27aba2495beea71c7b36758c9ef4da

How researchers see this paper

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In Computational Mechanics

1. Szymon Majewski
2. Gersende Fort
3. Yves F. Atchadé
4. Eric Moulines
5. Blazej Miasojedow
6. Francis R. Bach
7. Aymeric Dieuleveut
8. Sham M. Kakade

In Computational Theory and Mathematics

1. Gersende Fort
2. Szymon Majewski
3. Yves F. Atchadé
4. Eric Moulines
5. Blazej Miasojedow
6. Gareth O. Roberts
7. S. Jarner
8. Galin L. Jones

In Mathematical Physics

1. Gersende Fort
2. Eric Moulines
3. Yves F. Atchadé
4. Gareth O. Roberts
5. Galin L. Jones
6. S. Jarner
7. Hiroki Masuda
8. Arnaud Guillin

Citations in the corpus

• Analysis of nonsmooth stochastic approximation: the differential inclusion approach (2018, background): “case when f and g are both convex, f gradient Lipschitz and g lower semi-continuous; see for example [37, 32, 40, 18].”
• On Perturbed Proximal Gradient Algorithms (2014, background): “…bias is mitigated in the algorithm requires substantial mathematical developments , which are not covered by the results currently available in the literature (see e.g. Combettes and Pesquet (2015a); Rosasco et al. (2014); Combettes and Pesquet (2015b); Rosasco et al. (2015); Lin et al. (2015)).”