Convergence of Time-Averaged Mean Field Gradient Descent Dynamics for Continuous Multi-Player Zero-Sum Games
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915283784957952 |
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| author | Lu, Yulong Monmarché, Pierre |
| author_facet | Lu, Yulong Monmarché, Pierre |
| contents | The approximation of mixed Nash equilibria (MNE) for zero-sum games with mean-field interacting players has recently raised much interest in machine learning. In this paper we propose a mean-field gradient descent dynamics for finding the MNE of zero-sum games involving $K$ players with $K\geq 2$. The evolution of the players' strategy distributions follows coupled mean-field gradient descent flows with momentum, incorporating an exponentially discounted time-averaging of gradients. First, in the case of a fixed entropic regularization, we prove an exponential convergence rate for the mean-field dynamics to the mixed Nash equilibrium with respect to the total variation metric. This improves a previous polynomial convergence rate for a similar time-averaged dynamics with different averaging factors. Moreover, unlike previous two-scale approaches for finding the MNE, our approach treats all player types on the same time scale. We also show that with a suitable choice of decreasing temperature, a simulated annealing version of the mean-field dynamics converges to an MNE of the initial unregularized problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_07642 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Convergence of Time-Averaged Mean Field Gradient Descent Dynamics for Continuous Multi-Player Zero-Sum Games Lu, Yulong Monmarché, Pierre Optimization and Control Machine Learning Analysis of PDEs Probability 35Q89, 49N80, 91A16, 90C47 The approximation of mixed Nash equilibria (MNE) for zero-sum games with mean-field interacting players has recently raised much interest in machine learning. In this paper we propose a mean-field gradient descent dynamics for finding the MNE of zero-sum games involving $K$ players with $K\geq 2$. The evolution of the players' strategy distributions follows coupled mean-field gradient descent flows with momentum, incorporating an exponentially discounted time-averaging of gradients. First, in the case of a fixed entropic regularization, we prove an exponential convergence rate for the mean-field dynamics to the mixed Nash equilibrium with respect to the total variation metric. This improves a previous polynomial convergence rate for a similar time-averaged dynamics with different averaging factors. Moreover, unlike previous two-scale approaches for finding the MNE, our approach treats all player types on the same time scale. We also show that with a suitable choice of decreasing temperature, a simulated annealing version of the mean-field dynamics converges to an MNE of the initial unregularized problem. |
| title | Convergence of Time-Averaged Mean Field Gradient Descent Dynamics for Continuous Multi-Player Zero-Sum Games |
| topic | Optimization and Control Machine Learning Analysis of PDEs Probability 35Q89, 49N80, 91A16, 90C47 |
| url | https://arxiv.org/abs/2505.07642 |