No-Regret Generative Modeling via Parabolic Monge-Amp\`ere PDE

Nabarun Deb; Tengyuan Liang

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arXiv · 2504.09279

No-Regret Generative Modeling via Parabolic Monge-Amp\`ere PDE

Nabarun Deb·Tengyuan Liang

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Abstract

We introduce a novel generative modeling framework based on a discretized parabolic Monge-Amp\`{e}re 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-Amp\`{e}re 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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Nabarun Deb, Tengyuan Liang. 2025-04-12. No-Regret Generative Modeling via Parabolic Monge-Amp\`ere PDE. https://arxiv.org/abs/2504.09279

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