通过轨迹正则化连接平均场博弈与归一化流
Bridging Mean-Field Games and Normalizing Flows with Trajectory Regularization
AI总结：
本文揭示了平均场博弈（MFGs）与归一化流（NFs）的联系，将NF训练转化为求解MFG，通过基于轨迹的公式化准确求解高维MFGs并利用传输成本正则化提升泛化性能。
摘要：
平均场博弈（MFGs）是一种用于包含大量交互智能体系统的建模框架。它们在经济学、金融学和博弈论中具有应用。归一化流（NFs）是一类深度生成模型，通过使用可逆映射来计算数据似然，该映射通常使用神经网络进行参数化。它们对于密度建模和数据生成非常有用。尽管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 的训练情境化为求解 MFG，揭示了 MFGs 和 NFs 之间的联系。这是通过将 MFG 问题转化为智能体轨迹，并用流架构对所得 MFG 的离散化进行参数化来实现的。基于这种联系，我们探索了两个研究方向。首先，我们采用表达力强的 NF 架构来准确求解高维 MFGs，从而避开传统数值方法中的维度灾难。与其他深度学习方法相比，我们基于轨迹的公式化将连续性方程编码在神经网络中，从而更好地近似了种群动态。其次，我们用传输成本对 NFs 的训练进行正则化，并展示了其在控制模型 Lipschitz 界方面的有效性，从而带来更好的泛化性能。我们通过各种合成和现实生活数据集上的综合实验展示了数值结果。
英文摘要：
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, which is typically parameterized by using 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 neural network, resulting in a better approximation of the 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
