The pseudomonotone stochastic variational inequality problem: Analytical statements and stochastic extragradient schemes

Kannan, A.; Shanbhag, Uday V.

doi:10.1109/acc.2014.6859377

Cited by 13 publications

(13 citation statements)

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“…Next, under strict pseudo-monotonicity assumption, we prove convergence of the generated iterate to the solution of (SCVI) in an almost sure sense, extending the results of [15] to the block coordinate settings. When considering SCVIs with strongly pseudo-monotone mappings, we obtain a bound of the order O d k on the mean squared error, where k is the iteration number and d is the number of blocks extending results in [15] to the block coordinate regime. This result differs from the rate analysis of [8] for the SBMD algorithm for stochastic optimization problems with strongly convex objectives in three aspects: (a) While the SBMD method addresses the optimization regime, our rate result applies to the broader class of problems, i.e., SVCIs; (b) The assumption of strong pseudomonotonicity in our work is weaker than the strong monotonicity of the gradient mapping in [8]; (c) In contrast with the SBMD scheme where an averaging scheme with a constant stepsize rule is employed for addressing problem (SCOP) (cf.…”

Section: Introductionmentioning

confidence: 84%

“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Yousefian

Nedić

Shanbhag

2018

Set-Valued Var. Anal

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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.

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Section: Introductionmentioning

confidence: 84%

Section: Introductionmentioning

confidence: 99%

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

Yousefian

Nedić

Shanbhag

2018

Set-Valued Var. Anal

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“…We summarize much of the prior results in Table 1. We believe that this is amongst the first efforts to contend with this problem class but it is worth noting that subsequent to the conference version of this paper [45], there have been at least two papers that have considered the solution of stochastic pseudomonotone variational inequality problems. 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 * ).…”

Section: Stochastic Approximation Schemesmentioning

confidence: 99%

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

Kannan

Shanbhag

2019

Comput Optim Appl

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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.

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“…Since the SA methods need less computational costs and storages than the SAA methods, they have been extensively studied and applied to solve various practical stochastic problems arising in engineering and economics. Recently, SA methods have been applied to deal with the considered SVI in [15][16][17][18][19][20][21][22][23]. These methods mainly combine the SA techniques with some kinds of projection methods for deterministic VI.…”

Section: Introductionmentioning

confidence: 99%

An Infeasible Stochastic Approximation and Projection Algorithm for Stochastic Variational Inequalities

Zhang

Yang

et al. 2019

J Optim Theory Appl

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In this paper, we consider a stochastic variational inequality, in which the mapping involved is an expectation of a given random function. Inspired by the work of He (Appl Math Optim 35:69-76, 1997) and the extragradient method proposed by Iusem et al. (SIAM J Optim 29:175-206, 2019), we propose an infeasible projection algorithm with line search scheme, which can be viewed as a modification of the above-mentioned method of Iusem et al. In particular, in the correction step, we replace the projection by computing search direction and stepsize, that is, we need only one projection at each iteration, while the method of Iusem et al. requires two projections at each iteration. Moreover, we use dynamic sampled scheme with line search to cope with the absence of Lipschitz constant and choose the stepsize to be bounded away from zero and the direction to be a descent direction. In the process of stochastic approximation, we iteratively reduce the variance of a stochastic error. Under appropriate assumptions, we derive some properties related to convergence, convergence rate, and oracle complexity. In particular, compared with the method of Iusem et al., our method uses less projections and has the same iteration complexity, which, however, has a higher oracle complexity for a given tolerance in a finite dimensional space. Finally, we report some numerical experiments to show its efficiency.

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The pseudomonotone stochastic variational inequality problem: Analytical statements and stochastic extragradient schemes

Cited by 13 publications

References 14 publications

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

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

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

An Infeasible Stochastic Approximation and Projection Algorithm for Stochastic Variational Inequalities

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