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As a technical contribution, we derive a new Evolution Variational Inequality tailored to the parabolic Monge\u2013Amp\u00e8re PDE, connecting geometry, transportation cost and regret. Our framework accommodates nonlog-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.<\/jats:p>","DOI":"10.1214\/26-aos2635","type":"journal-article","created":{"date-parts":[[2026,10,2]],"date-time":"2026-10-02T12:48:33Z","timestamp":1790945313000},"page":"2260-2285","source":"Crossref","is-referenced-by-count":0,"title":["No-regret generative modeling via parabolic Monge\u2013Amp\u00e8re PDE"],"prefix":"10.1214","volume":"54","author":[{"given":"Nabarun","family":"Deb","sequence":"first","affiliation":[{"name":"Booth School of Business, University of Chicago"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Tengyuan","family":"Liang","sequence":"additional","affiliation":[{"name":"Booth School of Business, University of Chicago"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"108","reference":[{"key":"19","doi-asserted-by":"publisher","unstructured":"<small>Chen, G.<\/small> and <small>Teboulle, M.<\/small> (1993). 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