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

Jinlong Lei, Vinayak V. Shanbhag

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

This paper considers an N-player stochastic Nash game in which the i mathbf{th} player minimizes a composite objective f-{i}(x)+r-{i}(x-{i}), where f-{i} is expectation-valued and r-{i} 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 sample-size 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.

Original languageEnglish (US)
Title of host publication2018 IEEE Conference on Decision and Control, CDC 2018
PublisherInstitute of Electrical and Electronics Engineers Inc.
Pages3547-3552
Number of pages6
ISBN (Electronic)9781538613955
DOIs
StatePublished - Jan 18 2019
Event57th IEEE Conference on Decision and Control, CDC 2018 - Miami, United States
Duration: Dec 17 2018Dec 19 2018

Publication series

NameProceedings of the IEEE Conference on Decision and Control
Volume2018-December
ISSN (Print)0743-1546

Conference

Conference57th IEEE Conference on Decision and Control, CDC 2018
CountryUnited States
CityMiami
Period12/17/1812/19/18

Fingerprint

Sample Size
Linearly
Game
Gradient
Stochastic Gradient
Composite materials
Distributed Protocol
Optimal Rates
Overlay
Iterate
Batch
Monotonicity
Rate of Convergence
Composite
Converge
Minimise
Iteration
Computing
Evaluation

All Science Journal Classification (ASJC) codes

  • Control and Systems Engineering
  • Modeling and Simulation
  • Control and Optimization

Cite this

Lei, J., & Shanbhag, V. V. (2019). Linearly Convergent Variable Sample-Size Schemes for Stochastic Nash Games: Best-Response Schemes and Distributed Gradient-Response Schemes. In 2018 IEEE Conference on Decision and Control, CDC 2018 (pp. 3547-3552). [8618953] (Proceedings of the IEEE Conference on Decision and Control; Vol. 2018-December). Institute of Electrical and Electronics Engineers Inc.. https://doi.org/10.1109/CDC.2018.8618953
Lei, Jinlong ; Shanbhag, Vinayak V. / Linearly Convergent Variable Sample-Size Schemes for Stochastic Nash Games : Best-Response Schemes and Distributed Gradient-Response Schemes. 2018 IEEE Conference on Decision and Control, CDC 2018. Institute of Electrical and Electronics Engineers Inc., 2019. pp. 3547-3552 (Proceedings of the IEEE Conference on Decision and Control).
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Lei, J & Shanbhag, VV 2019, Linearly Convergent Variable Sample-Size Schemes for Stochastic Nash Games: Best-Response Schemes and Distributed Gradient-Response Schemes. in 2018 IEEE Conference on Decision and Control, CDC 2018., 8618953, Proceedings of the IEEE Conference on Decision and Control, vol. 2018-December, Institute of Electrical and Electronics Engineers Inc., pp. 3547-3552, 57th IEEE Conference on Decision and Control, CDC 2018, Miami, United States, 12/17/18. https://doi.org/10.1109/CDC.2018.8618953

Linearly Convergent Variable Sample-Size Schemes for Stochastic Nash Games : Best-Response Schemes and Distributed Gradient-Response Schemes. / Lei, Jinlong; Shanbhag, Vinayak V.

2018 IEEE Conference on Decision and Control, CDC 2018. Institute of Electrical and Electronics Engineers Inc., 2019. p. 3547-3552 8618953 (Proceedings of the IEEE Conference on Decision and Control; Vol. 2018-December).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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AB - This paper considers an N-player stochastic Nash game in which the i mathbf{th} player minimizes a composite objective f-{i}(x)+r-{i}(x-{i}), where f-{i} is expectation-valued and r-{i} 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 sample-size 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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Lei J, Shanbhag VV. Linearly Convergent Variable Sample-Size Schemes for Stochastic Nash Games: Best-Response Schemes and Distributed Gradient-Response Schemes. In 2018 IEEE Conference on Decision and Control, CDC 2018. Institute of Electrical and Electronics Engineers Inc. 2019. p. 3547-3552. 8618953. (Proceedings of the IEEE Conference on Decision and Control). https://doi.org/10.1109/CDC.2018.8618953