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Difference of Convex Programming in the Wasserstein Space with Applications to MMD Optimization

arXiv 2606.27767 ↗First posted 26 Jun 2026cs.LG math.OCcited by 0

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Abstract (arXiv)

Optimizing functionals over the space of probability measures is now ubiquitous in machine learning. A widely used approach is to perform the optimization directly over the Wasserstein space, but many objective functionals of practical interest are non-convex along Wasserstein geodesics, making the analysis of standard first-order methods challenging. In this work, we study a class of objectives over the Wasserstein space that admit a difference-of-convex (DC) decomposition and we lift the classical convex-concave procedure (CCCP) to this setting. Under smoothness and strong convexity assumptions on the convex components of the decomposition, we prove almost stationarity along the iterates of the resulting algorithm. Our main focus is on the Maximum Mean Discrepancy (MMD) and the Energy Distance (ED) functionals, for which we develop explicit Wasserstein DC decompositions, and establish local convergence of the scheme under mild assumptions. Empirically, we show that well-chosen DC decompositions yield faster and more stable convergence than Wasserstein gradient descent on these MMD objectives.

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References · 57

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38 more
  1. 2020Refining Deep Generative Models via Discriminator Gradient Flowcited by 45
  2. 2023Generative Sliced MMD Flows with Riesz Kernelscited by 37
  3. 2018Policy Optimization as Wasserstein Gradient Flowscited by 34
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  13. 2023Nonparametric Generative Modeling with Conditional Sliced-Wasserstein Flowscited by 15
  14. 2025Wasserstein Policy Optimizationcited by 13
  15. 2024Mirror and Preconditioned Gradient Descent in Wasserstein Spacecited by 12
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  28. 2025DDEQs: Distributional Deep Equilibrium Models through Wasserstein Gradient Flowscited by 4
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