No-Regret Generative Modeling via Parabolic Monge-Ampère PDE
This provides a novel theoretical approach for generative modeling and variational inference, though it appears incremental in integrating existing techniques.
The paper tackles generative modeling by introducing a framework based on a parabolic Monge-Ampère PDE, derived from the Sinkhorn algorithm, which converges to an optimal Brenier map for non-log-concave distributions.
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.