, Moreover, we write fl prox ?g (x) := prox ?g(s,·) (x ? 2??f ? (s, x)) and ?g ? := ?g ? (s, x ? 2??f ? (s, x)), for all x ? X. By assumption, the sequel and write u n := u n (t), v n := v n (s, t), w n := w n (s, t), h ? := h ? (s, ·), ?g := ?g(s, ·), ?f := ?f (s, ·), ? := ? n , prox ?f := prox ?f (s,·) , ?f ? := ?f ? (s, ·), z := z(t)
Smooth primal-dual coordinate descent algorithms for nonsmooth convex optimization, NIPS, pp.5854-5863, 2017. ,
On stochastic proximal gradient algorithms, 2014. ,
On perturbed proximal gradient algorithms, J. Mach. Learn. Res, vol.18, issue.1, pp.310-342, 2017. ,
Familles d'opérateurs maximaux monotones et mesurabilité, Ann. Mat. Pura Appl, vol.120, issue.1, pp.35-111, 1979. ,
Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward-backward splitting, and regularized gauss-seidel methods, Math. Program, vol.137, issue.1-2, pp.91-129, 2013. ,
URL : https://hal.archives-ouvertes.fr/hal-00790042
Differential inclusions, Grundlehren der Mathematischen Wissenschaften, vol.264 ,
URL : https://hal.archives-ouvertes.fr/hal-01037957
Set-valued maps and viability theory, 1984. ,
Poincaré's recurrence theorem for set-valued dynamical systems, Ann. Polon. Math, vol.54, issue.1, pp.85-91, 1991. ,
Scheduling in a random environment: stability and asymptotic optimality, IEEE/ACM Trans. Netw, vol.21, issue.1, pp.258-271, 2013. ,
URL : https://hal.archives-ouvertes.fr/hal-01121985
Modular proximal optimization for multidimensional total-variation regularization, 2014. ,
Legendre functions and the method of random bregman projections, J. Convex Anal, vol.4, issue.1, pp.27-67, 1997. ,
Strong conical hull intersection property, bounded linear regularity, Jameson's property (G), and error bounds in convex optimization, Math. Program, vol.86, issue.1, pp.135-160, 1999. ,
Convex analysis and monotone operator theory in Hilbert spaces, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 2011. ,
URL : https://hal.archives-ouvertes.fr/hal-01517477
Robust distributed consensus using total variation, IEEE Trans. Automat. Contr, vol.61, issue.6, pp.1550-1564, 2016. ,
DOI : 10.1109/tac.2015.2471755
URL : https://hal.archives-ouvertes.fr/hal-01259850
Dynamics of stochastic approximation algorithms, Séminaire de Probabilités, XXXIII, vol.1709, pp.1-68, 1999. ,
Asymptotic pseudotrajectories and chain recurrent flows, with applications, J. Dynam. Differential Equations, vol.8, issue.1, pp.141-176, 1996. ,
Stochastic approximation algorithms with constant step size whose average is cooperative, Ann. Appl. Probab, vol.9, issue.1, pp.216-241, 1999. ,
Stochastic approximations and differential inclusions, SIAM J. Control Optim, vol.44, issue.1, pp.328-348, 2005. ,
Stochastic approximations and differential inclusions, II. Applications. Math. Oper. Res, vol.31, issue.4, pp.673-695, 2006. ,
Ergodic properties of weak asymptotic pseudotrajectories for semiflows, J. Dynam. Differential Equations, vol.12, issue.3, pp.579-598, 2000. ,
Adaptive algorithms and stochastic approximations, Applications of Mathematics, vol.22, 1990. ,
Incremental proximal methods for large scale convex optimization, Math. Program, vol.129, issue.2, pp.163-195, 2011. ,
Ergodic convergence of a stochastic proximal point algorithm, SIAM J. Optim, vol.26, issue.4, pp.2235-2260, 2016. ,
Performance of a distributed stochastic approximation algorithm, IEEE Trans. Inf. Theory, vol.59, issue.11, pp.7405-7418, 2013. ,
Dynamical behavior of a stochastic Forward-Backward algorithm using random monotone operators, J. Optim. Theory Appl, vol.171, issue.1, pp.90-120, 2016. ,
URL : https://hal.archives-ouvertes.fr/hal-01183959
Building stochastic optimization algorithms with random monotone operators, EUCCO, 2016. ,
Convergence d'un algorithme du gradient proximal stochastique à pas constant et généralisation aux opérateurs monotones aléatoires, GRETSI, 2017. ,
A constant step Forward-Backward algorithm involving random maximal monotone operators, J. Convex Anal, 2019. ,
URL : https://hal.archives-ouvertes.fr/hal-01725134
Constant step stochastic approximations involving differential inclusions: Stability, long-run convergence and applications, 2019. ,
, Convergence of probability measures. Wiley Series in Probability and Statistics: Probability and Statistics, 1999.
From error bounds to the complexity of first-order descent methods for convex functions, Mathematical Programming, vol.165, issue.2, pp.471-507, 2017. ,
Stochastic approximation. A dynamical systems viewpoint, 2008. ,
Convex analysis and nonlinear optimization, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol.3 ,
Large-scale machine learning with stochastic gradient descent, COMPSTAT'2010, pp.177-186, 2010. ,
Optimization methods for large-scale machine learning, SIAM Review, vol.60, issue.2, pp.223-311, 2018. ,
Distributed optimization and statistical learning via the alternating direction method of multipliers, Machine Learning, vol.3, pp.1-122, 2011. ,
Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland mathematics studies, 1973. ,
Sparse and stable markowitz portfolios, Proc. Natl. Acad. Sci. USA, vol.106, pp.12267-12272, 2009. ,
Asymptotic convergence of nonlinear contraction semigroups in Hilbert space, J. Funct. Anal, vol.18, pp.15-26, 1975. ,
A limited memory algorithm for bound constrained optimization, SIAM J. Sci. Comput, vol.16, issue.5, pp.1190-1208, 1995. ,
Convex analysis and measurable multifunctions, Lecture Notes in Mathematics, vol.580, 1977. ,
An introduction to total variation for image analysis. Theoretical foundations and numerical methods for sparse recovery, vol.9, pp.263-340, 2010. ,
URL : https://hal.archives-ouvertes.fr/hal-00437581
A first-order primal-dual algorithm for convex problems with applications to imaging, J. Math. Imaging Vision, vol.40, issue.1, pp.120-145, 2011. ,
URL : https://hal.archives-ouvertes.fr/hal-00490826
Signal inpainting on graphs via total variation minimization, ICASSP, pp.8267-8271, 2014. ,
Approche de Douglas-Rachford aléatoire par blocs appliquée à la régression logistique parcimonieuse, GRETSI, pp.1-4, 2017. ,
Spectral graph theory, vol.92, 1997. ,
Iterative construction of the resolvent of a sum of maximal monotone operators, J. Convex Anal, vol.16, issue.4, pp.727-748, 2009. ,
Stochastic quasi-Fejér block-coordinate fixed point iterations with random sweeping, SIAM J. Optim, vol.25, issue.2, pp.1221-1248, 2015. ,
Stochastic approximations and perturbations in forwardbackward splitting for monotone operators, Pure Appl. Funct. Anal, vol.1, issue.1, pp.13-37, 2016. ,
URL : https://hal.archives-ouvertes.fr/hal-01380000
Stochastic forward-backward and primal-dual approximation algorithms with application to online image restoration, EUSIPCO, pp.1813-1817, 2016. ,
URL : https://hal.archives-ouvertes.fr/hal-01422154
A direct algorithm for 1D total variation denoising, IEEE Signal Process. Lett, vol.20, issue.11, pp.1054-1057, 2013. ,
URL : https://hal.archives-ouvertes.fr/hal-00675043
A primal-dual splitting method for convex optimization involving Lipschitzian, proximable and linear composite terms, J. Optim. Theory Appl, vol.158, issue.2, pp.460-479, 2013. ,
URL : https://hal.archives-ouvertes.fr/hal-00609728
Local extremes, runs, strings and multiresolution, Ann. Stat, pp.1-48, 2001. ,
Elements of topological dynamics, Mathematics and its Applications, vol.257, 1993. ,
, Stochastic Approximation with Decreasing Gain: Convergence and Asymptotic Theory. Unpublished Lecture Notes, 2000.
Bridging the gap between constant step size stochastic gradient descent and markov chains, 2017. ,
URL : https://hal.archives-ouvertes.fr/hal-01565514
On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators, Math. Program, vol.55, issue.1, pp.293-318, 1992. ,
Asymptotic behavior of ? p-based Laplacian regularization in semi-supervised learning, COLT, pp.879-906, 2016. ,
Stochastic approximations of set-valued dynamical systems: convergence with positive probability to an attractor, Math. Oper. Res, vol.35, issue.3, pp.624-640, 2010. ,
URL : https://hal.archives-ouvertes.fr/hal-00383277
Ergodic properties of weak asymptotic pseudotrajectories for set-valued dynamical systems, Stoch. Dyn, vol.13, issue.1, p.23, 2013. ,
URL : https://hal.archives-ouvertes.fr/hal-00555214
Asymptotic behavior of a Markovian stochastic algorithm with constant step, SIAM J. Control Optim, vol.37, issue.5, pp.1456-1482, 1999. ,
Markov chains with discontinuous drifts have differential inclusion limits, Perform. Eval, vol.69, issue.12, pp.623-642, 2012. ,
URL : https://hal.archives-ouvertes.fr/hal-00787999
Sur l'approximation, par éléments finis d'ordre un, et la résolution, par pénalisation-dualité d'une classe de problèmes de dirichlet non linéaires. Revue française d'automatique, informatique, recherche opérationnelle. Analyse numérique, vol.9, pp.41-76, 1975. ,
Network lasso: Clustering and optimization in large graphs, SIGKDD, pp.387-396, 2015. ,
The average principle for parabolic and elliptic differential equations and Markov processes with small diffusions, Theor. Probab. Appl, vol.8, pp.1-21, 1963. ,
Integrals, conditional expectations, and martingales of multivalued functions, J. Multivar. Anal, vol.7, issue.1, pp.149-182, 1977. ,
About properties of the mean value functional and of the continuous infimal convolution in stochastic convex analysis, pp.763-789, 1976. ,
Thèse présentée à l'Université de Clermont-Ferrand II pour obtenir le grade de Docteur ès Sciences Mathématiques, 1977. ,
Stochastic blockmodels: First steps, Soc. Networks, vol.5, issue.2, pp.109-137, 1983. ,
Optimal rates for total variation denoising, COLT, pp.1115-1146, 2016. ,
Reflection methods for user-friendly submodular optimization, NIPS, pp.1313-1321, 2013. ,
URL : https://hal.archives-ouvertes.fr/hal-00905258
A dynamic programming algorithm for the fused lasso and ? 0-segmentation, J. Comput. Graph. Stat, vol.22, issue.2, pp.246-260, 2013. ,
Linear convergence of gradient and proximal-gradient methods under the Polyak-?ojasiewicz condition, ECML-PKDD, pp.795-811, 2016. ,
Stochastic approximation and recursive algorithms and applications, Applications of Mathematics, vol.35, 2003. ,
, , 2014.
Cut pursuit: Fast algorithms to learn piecewise constant functions, AISTATS, pp.1384-1393, 2016. ,
URL : https://hal.archives-ouvertes.fr/hal-01306786
Approximation et optimisation, Collection Enseignement des Sciences, issue.13, 1972. ,
Gradient-based learning applied to document recognition, Proc. IEEE, vol.86, issue.11, pp.2278-2324, 1998. ,
Splitting algorithms for the sum of two nonlinear operators, SIAM J. Numer. Anal, vol.16, issue.6, pp.964-979, 1979. ,
Analysis of recursive stochastic algorithms, IEEE Trans. Automat. Contr, vol.22, issue.4, pp.551-575, 1977. ,
Analysis of nonsmooth stochastic approximation: the differential inclusion approach, 2018. ,
Locally adaptive regression splines, Ann. Stat, vol.25, issue.1, pp.387-413, 1997. ,
Brève communication. régularisation d'inéquations variationnelles par approximations successives. Revue française d'informatique et de recherche opérationnelle, Série rouge, vol.4, pp.154-158, 1970. ,
Mirror descent in saddle-point problems: Going the extra (gradient) mile, 2018. ,
URL : https://hal.archives-ouvertes.fr/hal-01891551
Invariant measures for set-valued dynamical systems, Trans. Amer. Math. Soc, vol.351, issue.3, pp.1203-1225, 1999. ,
Monotone (nonlinear) operators in Hilbert space, Duke Mathematical Journal, vol.29, issue.3, pp.341-346, 1962. ,
Theory of random sets. Probability and its Applications, 2005. ,
Fonctions convexes duales et points proximaux dans un espace Hilbertien, CR Acad. Sci. Paris Ser. A Math, vol.255, pp.2897-2899, 1965. ,
URL : https://hal.archives-ouvertes.fr/hal-01867195
Non-asymptotic analysis of stochastic approximation algorithms for machine learning, NIPS, pp.451-459, 2011. ,
URL : https://hal.archives-ouvertes.fr/hal-00608041
An adaptive distributed asynchronous algorithm with application to target localization, CAMSAP 2017, pp.1-5, 2017. ,
Randomized projection methods for convex feasibility problems: conditioning and convergence rates, 2018. ,
Bases mathématiques du calcul des probabilités, Masson et Cie, Éditeurs, 1964. ,
Stochastic alternating direction method of multipliers, ICML, pp.80-88, 2013. ,
The dfs fused lasso: Linear-time denoising over general graphs, J. Mach. Learn. Res, vol.18, issue.1, pp.6410-6445, 2017. ,
Ergodic convergence to a zero of the sum of monotone operators in Hilbert space, J. Math. Anal. Appl, vol.72, issue.2, pp.383-390, 1979. ,
Nonasymptotic convergence of stochastic proximal point algorithms for constrained convex optimization, J. Mach. Learn. Res, 2017. ,
Evolution equations for maximal monotone operators: asymptotic analysis in continuous and discrete time, J. Convex Anal, vol.17, issue.3-4, pp.1113-1163, 2010. ,
Lectures on maximal monotone operators, Extracta Math, vol.12, issue.3, pp.193-230, 1997. ,
Proximal stochastic methods for nonsmooth nonconvex finite-sum optimization, NIPS, pp.1145-1153, 2016. ,
A stochastic approximation method, Ann. Math. Stat, vol.22, issue.3, pp.400-407, 1951. ,
A convergence theorem for non negative almost supermartingales and some applications, Optimizing Methods in Statistics, pp.233-257, 1971. ,
Convex analysis, Princeton Mathematical Series, issue.28, 1970. ,
On the interchange of subdifferentiation and conditional expectations for convex functionals, Stochastics, vol.7, issue.3, pp.173-182, 1982. ,
Variational analysis, volume 317 of Grundlehren der Mathematischen Wissenschaften ,
, , 1998.
, Convergence of stochastic proximal gradient algorithm, 2014.
, Stochastic inertial primal-dual algorithms, 2015.
A stochastic inertial forward-backward splitting algorithm for multivariate monotone inclusions, Optimization, vol.65, issue.6, pp.1293-1314, 2016. ,
Stochastic approximations with constant step size and differential inclusions, SIAM J. Control Optim, vol.51, issue.1, pp.525-555, 2013. ,
Stochastic proximal iteration: a non-asymptotic improvement upon stochastic gradient descent ,
Snake: a stochastic proximal gradient algorithm for regularized problems over large graphs, CAp, 2017. ,
URL : https://hal.archives-ouvertes.fr/hal-02287871
A stochastic Douglas-Rachford algorithm with constant step size, 2017. ,
A constant step stochastic douglas-rachford algorithm with application to non separable regularizations, ICASSP, pp.2886-2890, 2018. ,
A splitting algorithm for minimization under stochastic linear constraints, ISMP, 2018. ,
Snake: a stochastic proximal gradient algorithm for regularized problems over large graphs, IEEE Trans. Automat. Contr, 2019. ,
URL : https://hal.archives-ouvertes.fr/hal-02287871
A stochastic proximal point algorithm for total variation regularization over large scale graphs, CDC, pp.4490-4495, 2016. ,
Group sparse regularization for deep neural networks, Neurocomputing, vol.241, pp.81-89, 2017. ,
Online and stochastic Douglas-Rachford splitting method for large scale machine learning, 2013. ,
Algorithms, graph theory, and linear equations in laplacian matrices, Proc. ICM, vol.4, pp.2698-2722, 2010. ,
DOI : 10.1142/9789814324359_0164
URL : http://www.cs.yale.edu/homes/spielman/PAPERS/icm10post.pdf
Nearly linear time algorithms for preconditioning and solving symmetric, diagonally dominant linear systems, SIAM J. Matrix Anal. Appl, vol.35, issue.3, pp.835-885, 2014. ,
DOI : 10.1137/090771430
URL : http://cs-www.cs.yale.edu/homes/spielman/PAPERS/77143.pdf
A fast and flexible algorithm for the graph-fused lasso, 2015. ,
Adaptive piecewise polynomial estimation via trend filtering, Ann. Stat, vol.42, issue.1, pp.285-323, 2014. ,
DOI : 10.1214/13-aos1189
URL : https://doi.org/10.1214/13-aos1189
Asymptotic and finite-sample properties of estimators based on stochastic gradients, Ann. Stat, vol.45, issue.4, pp.1694-1727, 2017. ,
Stable robbins-monro approximations through stochastic proximal updates, 2015. ,
Laplacian solvers and their algorithmic applications, Theor. Comput. Sci, vol.8, issue.1-2, pp.1-141, 2012. ,
A splitting algorithm for dual monotone inclusions involving cocoercive operators, Adv. Appl. Math, vol.38, issue.3, pp.667-681, 2013. ,
Stochastic programs with recourse. II: On the continuity of the objective, SIAM J. Appl. Math, vol.17, pp.98-103, 1969. ,
DOI : 10.1137/0115113
Incremental constraint projection methods for variational inequalities, Math. Program, vol.150, issue.2, pp.321-363, 2015. ,
DOI : 10.1007/s10107-014-0769-x
Trend filtering on graphs, J. Mach. Learn. Res, vol.17, issue.105, pp.1-41, 2016. ,
Stochastic recursive inclusions with non-additive iterate-dependent markov noise, Stochastics, vol.90, issue.3, pp.330-363, 2018. ,
DOI : 10.1080/17442508.2017.1353984
URL : http://arxiv.org/pdf/1607.04735
Combined group and exclusive sparsity for deep neural networks, ICML, pp.3958-3966, 2017. ,
Functional analysis, berlin, p.126, 1965. ,
Online convex optimization with stochastic constraints, NIPS, pp.1427-1437, 2017. ,
Efficient methods for overlapping group lasso, NIPS, pp.352-360, 2011. ,
Stochastic three-composite convex minimization, NIPS, pp.4329-4337, 2016. ,
Semi-supervised learning using gaussian fields and harmonic functions, ICML, pp.912-919, 2003. ,