No-Regret Generative Modeling via Parabolic Monge-Ampère PDE

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Abstract:

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.

Citation

Nabarun Deb, and Tengyuan Liang. 2025. “No-Regret Generative Modeling via Parabolic Monge-Ampère PDE.” The Annals of Statistics (forthcoming).

@article{DebLiang2025,
  title = {No-Regret Generative Modeling via Parabolic Monge-Amp\`ere PDE},
  author = {Deb, Nabarun and Liang, Tengyuan},
  journal = {The Annals of Statistics},
  year = {2025},
  note = {forthcoming},
  eprint = {2504.09279},
  archivePrefix = {arXiv},
  primaryClass = {stat.ML},
  url = {https://arxiv.org/abs/2504.09279},
}

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