Bridging Mean-Field Games and Normalizing Flows with Trajectory Regularization
Bridging Mean-Field Games and Normalizing Flows with Trajectory Regularization
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DOI：
10.1016/j.jcp.2023.112155
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发表时间：
2022-06
期刊：
影响因子：
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通讯作者：
Han Huang;Jiajia Yu;Jie Chen;Rongjie Lai
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文献类型：
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作者：
Han Huang;Jiajia Yu;Jie Chen;Rongjie Lai
平均场博弈 (MFG) 是具有大量交互代理的系统的建模框架。它们在经济学、金融学和博弈论中都有应用。归一化流 (NF) 是一系列深度生成模型，它通过使用通常由神经网络参数化的可逆映射来计算数据可能性。它们对于密度建模和数据生成很有用。虽然对这两种模型都进行了积极的研究，但很少有人注意到两者之间的关系。在这项工作中，我们通过将 NF 的训练背景化为解决 MFG 来阐明 MFG 和 NF 之间的联系。这是通过根据代理轨迹重新表述 MFG 问题并使用流架构对所得 MFG 的离散化进行参数化来实现的。以此联系，我们探索了两个研究方向。首先，我们采用富有表现力的 NF 架构来精确求解高维 MFG，避免了传统数值方法中的维数灾难。与其他深度学习方法相比，我们基于轨迹的公式对网络架构中的连续性方程进行编码，以更好地近似种群动态。其次，我们用运输成本规范 NF 的训练，并展示了控制模型 Lipschitz 界限的有效性，从而获得更好的泛化性能。我们通过对各种合成和现实数据集的综合实验展示了数值结果。
Mean-field games (MFGs) are a modeling framework for systems with a large number of interacting agents. They have applications in economics, finance, and game theory. Normalizing flows (NFs) are a family of deep generative models that compute data likelihoods by using an invertible mapping typically parameterized by neural networks. They are useful for density modeling and data generation. While active research has been conducted on both models, few noted the relationship between the two. In this work, we unravel the connections between MFGs and NFs by contextualizing the training of an NF as solving the MFG. This is achieved by reformulating the MFG problem in terms of agent trajectories and parameterizing a discretization of the resulting MFG with flow architectures. With this connection, we explore two research directions. First, we employ expressive NF architectures to accurately solve high-dimensional MFGs, sidestepping the curse of dimensionality in traditional numerical methods. Compared with other deep learning approaches, our trajectory-based formulation encodes the continuity equation in the network architecture to better approximate population dynamics. Second, we regularize the training of NFs with transport costs and show the effectiveness on controlling the model's Lipschitz bound, resulting in better generalization performance. We demonstrate numerical results through comprehensive experiments on a variety of synthetic and real-life datasets.