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Local Algebraic Stability of Wasserstein Gradient Descent-Ascent and Mean Field Langevin Dynamics without Spectral Gap

  • Jing An
  • Lihan Wang
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Abstract

We study the local stability of Wasserstein gradient descent-ascent (GDA) for entropy regularized two-player zero-sum games on noncompact domains. When one or both equilibrium measures fail to satisfy a Poincaré inequality, the spectral gap arguments underlying local exponential convergence are unavailable. we show that Wasserstein GDA converges algebraically to the mixed Nash equilibrium in chi-squared divergence, provided the initial relative density is a small perturbation in L2 with bounded algebraic moments. The proof has two main ingredients: a cancellation in a negative Sobolev energy estimate, and a bootstrap argument that keeps the moments bounded uniformly in time. For polynomial-tailed equilibria, we construct perturbations supported far out in the tails that survive over diffusive time scales, which gives algebraic lower bounds on the worst-case relaxation rate. We also establish analogous local algebraic stability and lower bound results for mean-field Langevin dynamics, when the equilibrium fails to satisfy a spectral gap.

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Cite

BibTeXRIS
BibTeX
@misc{an2026local,
  title = {{Local Algebraic Stability of Wasserstein Gradient Descent-Ascent and Mean Field Langevin Dynamics without Spectral Gap}},
  author = {Jing An and Lihan Wang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.35528},
  eprint = {2610.35528},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.35528},
}
RIS
TY  - UNPB
TI  - Local Algebraic Stability of Wasserstein Gradient Descent-Ascent and Mean Field Langevin Dynamics without Spectral Gap
AU  - Jing An
AU  - Lihan Wang
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.35528
UR  - https://arxiv.org/abs/2610.35528
AB  - We study the local stability of Wasserstein gradient descent-ascent (GDA) for entropy regularized two-player zero-sum games on noncompact domains. When one or both equilibrium measures fail to satisfy a Poincaré inequality, the spectral gap arguments underlying local exponential convergence are unavailable. we show that Wasserstein GDA converges algebraically to the mixed Nash equilibrium in chi-squared divergence, provided the initial relative density is a small perturbation in L^2 with bounded algebraic moments. The proof has two main ingredients: a cancellation in a negative Sobolev energy estimate, and a bootstrap argument that keeps the moments bounded uniformly in time. For polynomial-tailed equilibria, we construct perturbations supported far out in the tails that survive over diffusive time scales, which gives algebraic lower bounds on the worst-case relaxation rate. We also establish analogous local algebraic stability and lower bound results for mean-field Langevin dynamics, when the equilibrium fails to satisfy a spectral gap.
ER  -