Solving Variational Inequalities with Stochastic Mirror-Prox Algorithm

Abstract

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 Feasibility problem and deterministic Eigenvalue minimization.1. Introduction. Variational inequalities with monotone operators form a convenient framework for unified treatment (including algorithmic design) of problems with "convex structure", like convex minimization, convexconcave saddle point problems and convex Nash equilibrium problems. In this paper we utilize this framework to develop first order algorithms for stochastic versions of the outlined problems, where the precise first order information is replaced with its unbiased stochastic estimates. This situation arises naturally in convex Stochastic Programming, where the precise first order information is unavailable (see examples in section 4). In some situations, e.g. those considered in [4, Section 3.3] and in Section 4.4, where passing from available, but relatively computationally expensive precise first order information to its cheap stochastic estimates allows to accelerate the solution process, with the gain from randomization growing progressively with problem's sizes.Our "unifying framework" is as follows. Let Z be a convex compact set in Euclidean space E with inner product ·, · , · be a norm on E (not

Randomized first order algorithms with applications to ℓ 1-minimization

“…The deterministic s.p. prototypes of the randomized algorithms we develop here were proposed in [12,13], and the prototypes of our randomization scheme were proposed in [14,Section 3.3] and [10]. In this paper, we demonstrate that in the case of a bilinear s.p.…”

“…4, this feature of our algorithms becomes instrumental when solving GBSP problems. 4 This is in sharp contrast with the prototypes of the SA and the SMP proposed, respectively, in [14, Section 3.3] and [10]. The approximate solutions z t of those algorithms were computed according to the formula (28), but with z τ [14] or w τ [10] in the role of ζ τ .…”

“…4 This is in sharp contrast with the prototypes of the SA and the SMP proposed, respectively, in [14, Section 3.3] and [10]. The approximate solutions z t of those algorithms were computed according to the formula (28), but with z τ [14] or w τ [10] in the role of ζ τ . As a result, in the prototype algorithms there is no universal and computationally cheap way to quantify the quality of approximate solutions.…”

Almost sure convergence of the forward–backward–forward splitting algorithm

“…Furthermore, in the case when B is cocoercive and uniformly monotone, the almost sure convergence of the forward-backward splitting is also proved in [21] under different conditions on stepsize and stochastic errors. One of the early work concerns with Lipschitzian monotone operator was in [18]. Let (a n ) n∈N , (b n ) n∈N , and (c n ) n∈N be sequences of square integrable K-valued random vectors such that (3.33) and (3.34) are satisfied.…”

Subsampling algorithms for semidefinite programming

“…Strikingly, the algorithm has a total complexity of O(n log n/ǫ 2 ), when the problem matrix is n × n, hence only requires access to a negligible proportion of the matrix coefficients as the dimension n tends to infinity. A similar subsampling argument was used in [JNT08] to solve a variational inequality representation of maximum eigenvalue minimization problems.…”