Solving variational inequalities with Stochastic Mirror-Prox algorithm

In this paper we consider iterative methods for stochastic variational inequalities (s.v.i.) with monotone operators. Our basic assumption is that the operator possesses both smooth and nonsmooth components. Further, only noisy observations of the problem data are available. We develop a novel Stochastic Mirror-Prox (SMP) algorithm for solving s.v.i. and show that with the convenient stepsize strategy it attains the optimal rates of convergence with respect to the problem parameters. We apply the SMP algorithm to Stochastic composite minimization and describe particular applications to Stochastic Semidefinite Feasability problem and Eigenvalue minimization

Convex optimization over intersection of simple sets: improved convergence rate guarantees via an exact penalty approach

We consider the problem of minimizing a convex function over the intersection of finitely many simple sets which are easy to project onto. This is an important problem arising in various domains such as machine learning. The main difficulty lies in finding the projection of a point in the intersection of many sets. Existing approaches yield an infeasible point with an iteration-complexity of $O(1/\varepsilon^2)$ for nonsmooth problems with no guarantees on the in-feasibility. By reformulating the problem through exact penalty functions, we derive first-order algorithms which not only guarantees that the distance to the intersection is small but also improve the complexity to $O(1/\varepsilon)$ and $O(1/\sqrt{\varepsilon})$ for smooth functions. For composite and smooth problems, this is achieved through a saddle-point reformulation where the proximal operators required by the primal-dual algorithms can be computed in closed form. We illustrate the benefits of our approach on a graph transduction problem and on graph matching

Semi-proximal Mirror-Prox for Nonsmooth Composite Minimization

We propose a new first-order optimisation algorithm to solve high-dimensional non-smooth composite minimisation problems. Typical examples of such problems have an objective that decomposes into a non-smooth empirical risk part and a non-smooth regularisation penalty. The proposed algorithm, called Semi-Proximal Mirror-Prox, leverages the Fenchel-type representation of one part of the objective while handling the other part of the objective via linear minimization over the domain. The algorithm stands in contrast with more classical proximal gradient algorithms with smoothing, which require the computation of proximal operators at each iteration and can therefore be impractical for high-dimensional problems. We establish the theoretical convergence rate of Semi-Proximal Mirror-Prox, which exhibits the optimal complexity bounds, i.e. $O(1/\epsilon^2)$, for the number of calls to linear minimization oracle. We present promising experimental results showing the interest of the approach in comparison to competing methods

Decomposition in conic optimization with partially separable structure

Decomposition techniques for linear programming are difficult to extend to conic optimization problems with general non-polyhedral convex cones because the conic inequalities introduce an additional nonlinear coupling between the variables. However in many applications the convex cones have a partially separable structure that allows them to be characterized in terms of simpler lower-dimensional cones. The most important example is sparse semidefinite programming with a chordal sparsity pattern. Here partial separability derives from the clique decomposition theorems that characterize positive semidefinite and positive-semidefinite-completable matrices with chordal sparsity patterns. The paper describes a decomposition method that exploits partial separability in conic linear optimization. The method is based on Spingarn's method for equality constrained convex optimization, combined with a fast interior-point method for evaluating proximal operators

Sparse Non Gaussian Component Analysis by Semidefinite Programming

Sparse non-Gaussian component analysis (SNGCA) is an unsupervised method of extracting a linear structure from a high dimensional data based on estimating a low-dimensional non-Gaussian data component. In this paper we discuss a new approach to direct estimation of the projector on the target space based on semidefinite programming which improves the method sensitivity to a broad variety of deviations from normality. We also discuss the procedures which allows to recover the structure when its effective dimension is unknown