Computing Equilibria in Stochastic Nonconvex and Non-monotone Games via Gradient-Response Schemes
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913852933799936 |
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| author | Xiao, Zhuoyu Shanbhag, Uday V. |
| author_facet | Xiao, Zhuoyu Shanbhag, Uday V. |
| contents | We consider a class of smooth $N$-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 {\em quasi}-Nash equilibrium (QNE) via a single-step gradient-response framework. First, under a suitably defined quadratic growth property, we prove that the stochastic synchronous gradient-response ({\bf SSGR}) scheme and its asynchronous counterpart ({\bf 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 NE, rather than QNE. Second, under a weak sharpness property, we show that the deterministic synchronous variant displays a {\em 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. Third, when player problems are convex but the associated concatenated gradient map is potentially non-monotone, we prove that a zeroth-order asynchronous modified gradient-response ({\bf ZAMGR}) scheme can efficiently compute NE under a suitable copositivity requirement. Collectively, our findings represent amongst the first inroads into efficient computation of QNE/NE in nonconvex settings, leading to a set of single-step schemes that are characterized by broader reach while often providing last-iterate rate guarantees. We present applications satisfying the prescribed requirements where preliminary numerics appear promising. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_14056 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Computing Equilibria in Stochastic Nonconvex and Non-monotone Games via Gradient-Response Schemes Xiao, Zhuoyu Shanbhag, Uday V. Optimization and Control We consider a class of smooth $N$-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 {\em quasi}-Nash equilibrium (QNE) via a single-step gradient-response framework. First, under a suitably defined quadratic growth property, we prove that the stochastic synchronous gradient-response ({\bf SSGR}) scheme and its asynchronous counterpart ({\bf 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 NE, rather than QNE. Second, under a weak sharpness property, we show that the deterministic synchronous variant displays a {\em 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. Third, when player problems are convex but the associated concatenated gradient map is potentially non-monotone, we prove that a zeroth-order asynchronous modified gradient-response ({\bf ZAMGR}) scheme can efficiently compute NE under a suitable copositivity requirement. Collectively, our findings represent amongst the first inroads into efficient computation of QNE/NE in nonconvex settings, leading to a set of single-step schemes that are characterized by broader reach while often providing last-iterate rate guarantees. We present applications satisfying the prescribed requirements where preliminary numerics appear promising. |
| title | Computing Equilibria in Stochastic Nonconvex and Non-monotone Games via Gradient-Response Schemes |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2504.14056 |