Inexact best-response schemes for stochastic Nash games: Linear convergence and Iteration complexity analysis

Uday V. Shanbhag, Jong Shi Pang, Suvrajeet Sen

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

5 Citations (Scopus)

Abstract

We consider a subclass of N-player stochastic Nash games in which each player solves a parametrized stochastic optimization problem. In deterministic regimes, best response schemes have been shown to be convergent under a suitable spectral property associated with the proximal-response map. However, a direct application of this scheme to stochastic settings requires obtaining exact solutions to stochastic optimization problems at every step. Instead, we propose an inexact generalization of this scheme in which an inexact solution to the best response problem is computed where the player-specific inexactness sequence is assumed to be separable. Notably, this scheme is an implementable single-loop scheme that requires a fixed (but increasing) number of stochastic gradient steps to compute an inexact solution to the best response problem. On the basis of this framework, we make several contributions: (i) The presented inexact best-response scheme produces iterates that converge to the unique equilibrium in mean; (ii) Surprisingly, we show that the iterates converge at a prescribed linear rate with a prescribed constant rather than a sub-linear rate; and (iii) Finally, by assuming that an inexact solution is computed by a stochastic approximation scheme, the overall iteration complexity for computing an -Nash equilibrium less that O(√N/)2+δ where δ is a positive scalar. Additionally, we show that the upper bound of this effort is shown to be N=2).

Original languageEnglish (US)
Title of host publication2016 IEEE 55th Conference on Decision and Control, CDC 2016
PublisherInstitute of Electrical and Electronics Engineers Inc.
Pages3591-3596
Number of pages6
ISBN (Electronic)9781509018376
DOIs
StatePublished - Dec 27 2016
Event55th IEEE Conference on Decision and Control, CDC 2016 - Las Vegas, United States
Duration: Dec 12 2016Dec 14 2016

Publication series

Name2016 IEEE 55th Conference on Decision and Control, CDC 2016

Other

Other55th IEEE Conference on Decision and Control, CDC 2016
CountryUnited States
CityLas Vegas
Period12/12/1612/14/16

Fingerprint

Linear Convergence
Complexity Analysis
Game
Iteration
Stochastic Optimization
Iterate
Optimization Problem
Converge
Stochastic Gradient
Stochastic Approximation
Approximation Scheme
Spectral Properties
Nash Equilibrium
Best response
Exact Solution
Scalar
Upper bound
Computing
Stochastic optimization
Optimization problem

All Science Journal Classification (ASJC) codes

  • Artificial Intelligence
  • Decision Sciences (miscellaneous)
  • Control and Optimization

Cite this

Shanbhag, U. V., Pang, J. S., & Sen, S. (2016). Inexact best-response schemes for stochastic Nash games: Linear convergence and Iteration complexity analysis. In 2016 IEEE 55th Conference on Decision and Control, CDC 2016 (pp. 3591-3596). [7798809] (2016 IEEE 55th Conference on Decision and Control, CDC 2016). Institute of Electrical and Electronics Engineers Inc.. https://doi.org/10.1109/CDC.2016.7798809
Shanbhag, Uday V. ; Pang, Jong Shi ; Sen, Suvrajeet. / Inexact best-response schemes for stochastic Nash games : Linear convergence and Iteration complexity analysis. 2016 IEEE 55th Conference on Decision and Control, CDC 2016. Institute of Electrical and Electronics Engineers Inc., 2016. pp. 3591-3596 (2016 IEEE 55th Conference on Decision and Control, CDC 2016).
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Shanbhag, UV, Pang, JS & Sen, S 2016, Inexact best-response schemes for stochastic Nash games: Linear convergence and Iteration complexity analysis. in 2016 IEEE 55th Conference on Decision and Control, CDC 2016., 7798809, 2016 IEEE 55th Conference on Decision and Control, CDC 2016, Institute of Electrical and Electronics Engineers Inc., pp. 3591-3596, 55th IEEE Conference on Decision and Control, CDC 2016, Las Vegas, United States, 12/12/16. https://doi.org/10.1109/CDC.2016.7798809

Inexact best-response schemes for stochastic Nash games : Linear convergence and Iteration complexity analysis. / Shanbhag, Uday V.; Pang, Jong Shi; Sen, Suvrajeet.

2016 IEEE 55th Conference on Decision and Control, CDC 2016. Institute of Electrical and Electronics Engineers Inc., 2016. p. 3591-3596 7798809 (2016 IEEE 55th Conference on Decision and Control, CDC 2016).

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

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AB - We consider a subclass of N-player stochastic Nash games in which each player solves a parametrized stochastic optimization problem. In deterministic regimes, best response schemes have been shown to be convergent under a suitable spectral property associated with the proximal-response map. However, a direct application of this scheme to stochastic settings requires obtaining exact solutions to stochastic optimization problems at every step. Instead, we propose an inexact generalization of this scheme in which an inexact solution to the best response problem is computed where the player-specific inexactness sequence is assumed to be separable. Notably, this scheme is an implementable single-loop scheme that requires a fixed (but increasing) number of stochastic gradient steps to compute an inexact solution to the best response problem. On the basis of this framework, we make several contributions: (i) The presented inexact best-response scheme produces iterates that converge to the unique equilibrium in mean; (ii) Surprisingly, we show that the iterates converge at a prescribed linear rate with a prescribed constant rather than a sub-linear rate; and (iii) Finally, by assuming that an inexact solution is computed by a stochastic approximation scheme, the overall iteration complexity for computing an -Nash equilibrium less that O(√N/)2+δ where δ is a positive scalar. Additionally, we show that the upper bound of this effort is shown to be N=2).

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Shanbhag UV, Pang JS, Sen S. Inexact best-response schemes for stochastic Nash games: Linear convergence and Iteration complexity analysis. In 2016 IEEE 55th Conference on Decision and Control, CDC 2016. Institute of Electrical and Electronics Engineers Inc. 2016. p. 3591-3596. 7798809. (2016 IEEE 55th Conference on Decision and Control, CDC 2016). https://doi.org/10.1109/CDC.2016.7798809