Regularized Iterative Stochastic Approximation Methods for Stochastic Variational Inequality Problems

Koshal, Jayash; Nedić, Angelia; Shanbhag, Uday V.

doi:10.1109/tac.2012.2215413

Cited by 149 publications

(188 citation statements)

References 34 publications

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“…The reader is referred to the recent publications [13] and [17] for a discussion of the challenges associated with this problem and its applications (our use of the "≤" relation instead of the common "≥" is only motivated by the easiness to show the conversion to our formulation). We propose to convert problem (1.3) to the nested form (1.1) by defining the lifted gap function f : Ê n × Ê n → Ê as 4) and the function g : Ê n → Ê n × Ê n as g(x) = x, [H(x)] .…”

Section: Example 1 (Stochastic Variational Inequality)mentioning

confidence: 99%

A Single Timescale Stochastic Approximation Method for Nested Stochastic Optimization

2020

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We study constrained nested stochastic optimization problems in which the objective function is a composition of two smooth functions whose exact values and derivatives are not available. We propose a single time-scale stochastic approximation algorithm, which we call the Nested Averaged Stochastic Approximation (NASA), to find an approximate stationary point of the problem. The algorithm has two auxiliary averaged sequences (filters) which estimate the gradient of the composite objective function and the inner function value. By using a special Lyapunov function, we show that NASA achieves the sample complexity of O(1/ε 2 ) for finding an ε-approximate stationary point, thus outperforming all extant methods for nested stochastic approximation. Our method and its analysis are the same for both unconstrained and constrained problems, without any need of batch samples for constrained nonconvex stochastic optimization. We also present a simplified parameter-free variant of the NASA method for solving constrained single level stochastic optimization problems, and we prove the same complexity result for both unconstrained and constrained problems.

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Section: Example 1 (Stochastic Variational Inequality)mentioning

confidence: 99%

A Single Timescale Stochastic Approximation Method for Nested Stochastic Optimization

2020

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“…Note that F k refers to the σ-field associated with k and ω k . Unfortunately, the almost-sure convergence statements of standard stochastic approximation schemes are limited to regimes where the map is strongly monotone while regularized variants have shown capable of solving merely monotone or strictly monotone maps [11], [12]. Motivated by this shortcoming, we propose what we believe is the first stochastic extragradient scheme.…”

Section: A Stochastic Extragradient Frameworkmentioning

confidence: 99%

“…Of these, the first approximates expectations by sample averages and examines the consistency of the resulting estimators (sample average approximation (SAA) techniques) [8], [9]. The second approach, inspired by the seminal work by Robbins and Monro [10], is that of stochastic approximation and there has been a surge of recent effort applying such techniques to stochastic variational inequality problems [11], [12], [13], [14]. While most of the above contributions rely on standard diminishing steplength sequences, Yousefian and his coauthors [15], [16] propose techniques in which the steplength sequences are tuned to problem parameters and minimize a suitably defined meansquared error bound.…”

Section: Introductionmentioning

confidence: 99%

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

Kannan

Shanbhag

2014

2014 American Control Conference

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Variational inequality problems find wide applicability in modeling a range of optimization and equilibrium problems. We consider the stochastic generalization of such a problem wherein the mapping is pseudomonotone and make two sets of contributions in this paper. First, we provide sufficiency conditions for the solvability of such problems that do not require evaluating the expectation. Second, we consider an extragradient variant of stochastic approximation for the solution of such problems and under suitable conditions, show that this scheme produces iterates that converge in an almostsure sense.

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“…(ii) Stochastic Nash games. Regularized stochastic approximation schemes were presented for monotone stochastic Nash games [8] while extensions have been developed to contend with misspecification [9] and the lack of Lipschitzian properties [10]. More recently, sampled bestresponse schemes have been developed in [11] while rate statements and iteration complexity bounds have been provided for a class of inexact stochastic best-response schemes in [12]- [14].…”

Section: Introductionmentioning

confidence: 99%

Linearly Convergent Variable Sample-Size Schemes for Stochastic Nash Games: Best-Response Schemes and Distributed Gradient-Response Schemes

Lei

Shanbhag

2018

2018 IEEE Conference on Decision and Control (CDC)

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This paper considers an N -player stochastic Nash game in which the ith player minimizes a composite objective fi(x) + ri(xi), where fi is expectation-valued and ri has an efficient prox-evaluation. In this context, we make the following contributions. (i) Under a strong monotonicity assumption on the concatenated gradient map, we derive (optimal) rate statements and oracle complexity bounds for the proposed variable sample-size proximal stochastic gradient-response (VS-PGR) scheme; (ii) We overlay (VS-PGR) with a consensus phase with a view towards developing distributed protocols for aggregative stochastic Nash games. Notably, when the samplesize and the number of consensus steps at each iteration grow at a suitable rate, a linear rate of convergence can be achieved; (iii) Finally, under a suitable contractive property associated with the proximal best-response (BR) map, we design a variable sample-size proximal BR (VS-PBR) scheme, where the proximal BR is computed by solving a sample-average problem. If the batch-size for computing the sample-average is raised at a suitable rate, we show that the resulting iterates converge at a linear rate and derive the oracle complexity.

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Regularized Iterative Stochastic Approximation Methods for Stochastic Variational Inequality Problems

Cited by 149 publications

References 34 publications

A Single Timescale Stochastic Approximation Method for Nested Stochastic Optimization

A Single Timescale Stochastic Approximation Method for Nested Stochastic Optimization

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

Linearly Convergent Variable Sample-Size Schemes for Stochastic Nash Games: Best-Response Schemes and Distributed Gradient-Response Schemes

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