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

Abstract

We consider the stochastic variational inequality problem in which the map is expectationvalued in a component-wise sense. Much of the available convergence theory and rate statements for stochastic approximation schemes is limited to monotone maps. However, non-monotone stochastic variational inequality problems are not uncommon and are seen to arise from product pricing, fractional optimization problems, and subclasses of economic equilibrium problems. Motivated by the need to address a broader class of maps, we make the following contributions: (i) We present an extragradient-based stochastic approximation scheme and prove that the iterates converge to a solution of the original problem under either pseudomonotonicity requirements or a suitably defined acute angle condition. Such statements are shown to be generalizable to the stochastic mirror-prox framework; (ii) Under strong pseudomonotonicity, we show that the mean-squared error in the solution iterates produced by the extragradient SA scheme converges at the optimal rate of O 1 K , statements that were hitherto unavailable in this regime. Notably, we optimize the initial steplength by obtaining an -infimum of a discontinuous nonconvex function. Similar statements are derived for mirror-prox generalizations and can accommodate monotone SVIs under a weak-sharpness requirement. Finally, both the asymptotics and the empirical rates of the schemes are studied on a set of variational problems where it is seen that the theoretically specified initial steplength leads to significant performance benefits.

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

“…where in the second inequality, we applied Jensen's inequality, and in the last inequality we used Assumption 2. Taking expectations in relation (14) with respect toξ k , and using the preceding estimate, we obtain…”

“…It is shown that under an averaging scheme, the SMP method generates iterates that converge to a weak solution of the stochastic VI. Kannan and Shanbhag (see [14,15]) studied almost sure convergence of extragradient algorithms in solving stochastic VIs with pseudo-monotone mappings and derived optimal rate statements under a strong pseudo-monotone condition. Recently, Iusem et al [10] developed an extragradient method with variance reduction for solving stochastic variational inequalities requiring only pseudo-monotonicity.…”

An Infeasible Stochastic Approximation and Projection Algorithm for Stochastic Variational Inequalities

“…In order to accurately compare the numerical effects of two algorithms, we chose one test problem randomly and drew the curve of y(k) := x k IPSA − x k ESA for comparison The second follows from [30]. For this example, we chose all initial point to be x 0 = 2e with suitable dimensions.…”

On the Convergence of (Stochastic) Gradient Descent with Extrapolation for Non-Convex Minimization

“…In the theoretical side, their convergence rates of proposed algorithms are built based on an assumption that the considered problems are convex-concave or the variational inequalities are monotone. Few works have considered (stochastic) extragradient methods for non-monotone VI (Kannan and Shanbhag, 2014;Dang and Lan, 2015) under some pseudo-monotonicity assumption. In contrast, we directly analyze GDE methods and their convergence for finding a stationary point of smooth nonconvex optimization problems without considering the above assumptions.…”