Computing Quasi-Nash Equilibria via Gradient-Response Schemes

Zhuoyu Xiao, Uday V. Shanbhag
Proceedings of the 7th Annual Learning for Dynamics \& Control Conference, PMLR 283:881-893, 2025.

Abstract

We consider a class of smooth static N-player noncooperative games, where player objectives are expectation-valued and potentially nonconvex. In such a setting, we consider the largely open question of efficiently computing a suitably defined quasi-Nash equilibrium (QNE) via a stochastic gradient-response framework. First, under a suitably defined quadratic growth property, we prove that both the stochastic synchronous gradient-response (SSGR) scheme and its asynchronous counterpart (SAGR) are characterized by almost sure convergence to a QNE and a sublinear rate guarantee. Notably, when a potentiality requirement is overlaid under a somewhat stronger pseudomonotonicity condition, this claim can be made for a Nash equilibrium (NE), rather than a QNE. Second, under the weak sharpness property, we show that the deterministic synchronous variant (SGR) displays a linear rate of convergence sufficiently close to a QNE by leveraging a geometric decay in steplengths. This suggests the development of a two-stage scheme with global non-asymptotic sublinear rates and a local linear rate. We also present applications satisfying the prescribed requirements where preliminary numerics appear promising.

Cite this Paper

BibTeX
@InProceedings{pmlr-v283-xiao25a, title = {Computing Quasi-Nash Equilibria via Gradient-Response Schemes}, author = {Xiao, Zhuoyu and Shanbhag, Uday V.}, booktitle = {Proceedings of the 7th Annual Learning for Dynamics \& Control Conference}, pages = {881--893}, year = {2025}, editor = {Ozay, Necmiye and Balzano, Laura and Panagou, Dimitra and Abate, Alessandro}, volume = {283}, series = {Proceedings of Machine Learning Research}, month = {04--06 Jun}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v283/main/assets/xiao25a/xiao25a.pdf}, url = {https://proceedings.mlr.press/v283/xiao25a.html}, abstract = {We consider a class of smooth static N-player noncooperative games, where player objectives are expectation-valued and potentially nonconvex. In such a setting, we consider the largely open question of efficiently computing a suitably defined quasi-Nash equilibrium (QNE) via a stochastic gradient-response framework. First, under a suitably defined quadratic growth property, we prove that both the stochastic synchronous gradient-response (SSGR) scheme and its asynchronous counterpart (SAGR) are characterized by almost sure convergence to a QNE and a sublinear rate guarantee. Notably, when a potentiality requirement is overlaid under a somewhat stronger pseudomonotonicity condition, this claim can be made for a Nash equilibrium (NE), rather than a QNE. Second, under the weak sharpness property, we show that the deterministic synchronous variant (SGR) displays a linear rate of convergence sufficiently close to a QNE by leveraging a geometric decay in steplengths. This suggests the development of a two-stage scheme with global non-asymptotic sublinear rates and a local linear rate. We also present applications satisfying the prescribed requirements where preliminary numerics appear promising.} }
Endnote
%0 Conference Paper %T Computing Quasi-Nash Equilibria via Gradient-Response Schemes %A Zhuoyu Xiao %A Uday V. Shanbhag %B Proceedings of the 7th Annual Learning for Dynamics \& Control Conference %C Proceedings of Machine Learning Research %D 2025 %E Necmiye Ozay %E Laura Balzano %E Dimitra Panagou %E Alessandro Abate %F pmlr-v283-xiao25a %I PMLR %P 881--893 %U https://proceedings.mlr.press/v283/xiao25a.html %V 283 %X We consider a class of smooth static N-player noncooperative games, where player objectives are expectation-valued and potentially nonconvex. In such a setting, we consider the largely open question of efficiently computing a suitably defined quasi-Nash equilibrium (QNE) via a stochastic gradient-response framework. First, under a suitably defined quadratic growth property, we prove that both the stochastic synchronous gradient-response (SSGR) scheme and its asynchronous counterpart (SAGR) are characterized by almost sure convergence to a QNE and a sublinear rate guarantee. Notably, when a potentiality requirement is overlaid under a somewhat stronger pseudomonotonicity condition, this claim can be made for a Nash equilibrium (NE), rather than a QNE. Second, under the weak sharpness property, we show that the deterministic synchronous variant (SGR) displays a linear rate of convergence sufficiently close to a QNE by leveraging a geometric decay in steplengths. This suggests the development of a two-stage scheme with global non-asymptotic sublinear rates and a local linear rate. We also present applications satisfying the prescribed requirements where preliminary numerics appear promising.
APA
Xiao, Z. & Shanbhag, U.V.. (2025). Computing Quasi-Nash Equilibria via Gradient-Response Schemes. Proceedings of the 7th Annual Learning for Dynamics \& Control Conference, in Proceedings of Machine Learning Research 283:881-893 Available from https://proceedings.mlr.press/v283/xiao25a.html.

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