用抛物型蒙日-安培方程构建生成模型，实现无遗憾收敛。
No-Regret Generative Modeling via Parabolic Monge-Ampère PDE
- 基于最优传输的镜像梯度下降迭代优化布莱尼尔映射。
- 在多种步长策略下可收敛到最优映射，支持非对数凹分布。
- 融合GAN与扩散模型优势，适合高维生成与变分推断任务。
我们提出一种基于离散抛物型蒙日-安培偏微分方程的新型生成建模框架，该方程是常用最优传输算法Sinkhorn的连续极限。方法通过镜像梯度下降在布莱尼尔映射空间中进行迭代优化，建立无遗憾分析理论，证明在多种步长调度下迭代序列收敛至最优布莱尼尔映射。技术上，我们推导出适用于抛物型蒙日-安培方程的新演化变分不等式，连接几何、运输成本与遗憾值。框架可处理非对数凹目标分布，通过布莱尼尔映射构建最优采样过程，并整合生成对抗网络与基于得分的扩散模型中的优良学习机制。作为直接应用，我们的理论为生成建模与变分推断开辟新路径。
原文摘要 · Abstract (English)
We introduce a novel generative modeling framework based on a discretized parabolic Monge-Ampère PDE, which emerges as a continuous limit of the Sinkhorn algorithm commonly used in optimal transport. Our method performs iterative refinement in the space of Brenier maps using a mirror gradient descent step. We establish theoretical guarantees for generative modeling through the lens of no-regret analysis, demonstrating that the iterates converge to the optimal Brenier map under a variety of step-size schedules. As a technical contribution, we derive a new Evolution Variational Inequality tailored to the parabolic Monge-Ampère PDE, connecting geometry, transportation cost, and regret. Our framework accommodates non-log-concave target distributions, constructs an optimal sampling process via the Brenier map, and integrates favorable learning techniques from generative adversarial networks and score-based diffusion models. As direct applications, we illustrate how our theory paves new pathways for generative modeling and variational inference.
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