Solving variational inequalities with Stochastic Mirror-Prox algorithm

Anatoli B. Juditsky (1) , Arkadii S. Nemirovski (2) , Claire Tauvel (1)

1 SAM - Statistique Apprentissage Machine
2 ISyE - School of Industrial and Systems Engineering [Georgia Tech]

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.

Mots clés: Variational inequalities with monotone operators; stochastic convex-concave saddle-point problem; large scale stochastic approximation; reduced complexity algorithms for convex optimization

Domaines: Optimisation et contrôle [math.OC]; Statistiques [math.ST]; Théorie [stat.TH]

Anatoli B. Juditsky, Arkadii S. Nemirovski, Claire Tauvel. Solving variational inequalities with Stochastic Mirror-Prox algorithm. Stochastic Systems, 2011, 1 (1), pp.17-58. ⟨10.1214/10-SSY011⟩. ⟨hal-00318043v2⟩