A first-order stochastic primal-dual algorithm with correction step

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 {\em Y. Drori, S. Sabach, and M. Teboulle, A simple algorithm for a class of nonsmooth convex-concave saddle-point problems, 2015} and {\em I. Loris and C. Verhoeven, On a generalization of the iterative soft-thresholding algorithm for the case of non-separable penalty, 2011} 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 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 Fej\'er-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 stochastic inertial forward-backward splitting algorithm for multivariate monotone inclusions

We propose an inertial forward-backward splitting algorithm to compute the 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

Structured Sparsity: Discrete and Convex approaches

Compressive sensing (CS) exploits sparsity to recover sparse or compressible signals from dimensionality reducing, non-adaptive sensing mechanisms. Sparsity is also used to enhance interpretability in machine learning and statistics applications: While the ambient dimension is vast in modern data analysis problems, the relevant information therein typically resides in a much lower dimensional space. However, many solutions proposed nowadays do not leverage the true underlying structure. Recent results in CS extend the simple sparsity idea to more sophisticated {\em structured} sparsity models, which describe the interdependency between the nonzero components of a signal, allowing to increase the interpretability of the results and lead to better recovery performance. In order to better understand the impact of structured sparsity, in this chapter we analyze the connections between the discrete models and their convex relaxations, highlighting their relative advantages. We start with the general group sparse model and then elaborate on two important special cases: the dispersive and the hierarchical models. For each, we present the models in their discrete nature, discuss how to solve the ensuing discrete problems and then describe convex relaxations. We also consider more general structures as defined by set functions and present their convex proxies. Further, we discuss efficient optimization solutions for structured sparsity problems and illustrate structured sparsity in action via three applications.Comment: 30 pages, 18 figure