On Stochastic Mirror-prox Algorithms for Stochastic Cartesian Variational Inequalities: Randomized Block Coordinate and Optimal Averaging Schemes

Abstract

Motivated by multi-user optimization problems and non-cooperative Nash games in uncertain regimes, we consider stochastic Cartesian variational inequality problems (SCVI) where the set is given as the Cartesian product of a collection of component sets. First, we consider the case where the number of the component sets is large. For solving this type of problems the classical stochastic approximation methods and their prox generalizations are computationally inefficient as each iteration becomes costly. To address this challenge, we develop a randomized block stochastic mirror-prox (B-SMP) algorithm, where at each iteration only a randomly selected block coordinate of the solution vector is updated through implementing two consecutive projection steps. Under standard assumptions on the problem and settings of the algorithm, we show that when the mapping is strictly pseudo-monotone, the algorithm generates a sequence of iterates that converges to the solution of the problem almost surely. To derive rate statements, we assume that the maps are strongly pseudo-monotone and obtain a non-asymptotic rate of O d k in mean-squared error, where k is the iteration number and d is the number of component sets. Second, we consider large-scale stochastic optimization problems with convex objectives. For this class of problems, we develop a new averaging scheme for the B-SMP algorithm. Unlike the classical averaging stochastic mirror-prox (SMP) method where a decreasing set of weights for the averaging sequence is used we consider a different set of weights that are characterized in terms of the stepsizes. We show that by using such weights, the objective values of the averaged sequence converges to the optimal value in the mean sense at the rate O √ d √ k . Both of the rate results appear to be new in the context of SMP algorithms. Third, we consider SCVIs and develop an SMP algorithm that employs the new weighted averaging scheme. We show that the expected value of a suitably defined gap function converges to zero at the optimal rate O 1 √ k , extending the previous rate results of the SMP algorithm.

Optimal stochastic extragradient schemes for pseudomonotone stochastic variational inequality problems and their variants

“…Of these, the first 1 utilizes a similar extragradient scheme with a.s. and rate statements that incorporates variable batch size [46], leading to an improved rate of O( 1 K ) in terms of dist 2 (x K , X * ). In addition, the second author has also recently jointly coauthored work on a block-coordinate variant of such schemes that incorporates a novel averaging scheme in the context of stochastic mirror-prox schemes [47].…”

A Randomized Block Coordinate Iterative Regularized Subgradient Method for High-dimensional Ill-posed Convex Optimization

“…Perhaps this interest lies in the strong interplay between the VIs and the formulation of optimization and equilibrium problems arising in many communication and networking problems [38]. Korpelevich's celebrated extragradient method [24] and its extensions [18,7,15,8,14,49] were developed which require weaker assumptions than their gradient counterparts. In the past decade, there has been a trending interest in addressing VIs in the stochastic regimes.…”

The Scenario Approach Meets Uncertain Game Theory and Variational Inequalities

“…Second claim: In parallel to the result of Proposition 2, it is not difficult to prove that, under the given assumptions, x SR is a sampled robust equilibrium if and only if it satisfies the variational inequality F (x SR ) ⊤ (x − x SR ) ≥ 0, ∀x ∈ X 1 × ... × X M , where F is given in (10). This can be shown by extending the result of [1, Prop.…”

“…Naturally, if the expectation can be easily evaluated, solving (2) is no harder than solving a deterministic variational inequality, for which much is known (e.g., existence and uniqueness results, algorithms [1]). If this is not the case, one could employ sampling-based algorithms to compute an approximate solution of (2), see [8]- [10]. A second approach, which we refer to as the robust formulation, is used to accommodate uncertainty both in the operator, and in the constraint sets.…”