Abstract
We study stability and sample complexity properties of divergence regularized optimal transport (DOT). First, we obtain quantitative stability results for optimizers of DOT measured in Wasserstein distance, which are applicable to a wide class of divergences and simultaneously improve known results for entropic optimal transport. Second, we study the case of sample complexity, where the DOT problem is approximated using empirical measures of the marginals. We show that divergence regularization can improve the corresponding convergence rate compared to unregularized optimal transport. To this end, we prove upper bounds which exploit both the regularity of cost function and divergence functional, as well as the intrinsic dimension of the marginals. Along the way, we establish regularity properties of dual optimizers of DOT, as well as general limit theorems for empirical measures with suitable classes of test functions.
Funding Statement
Erhan Bayraktar is supported in part by the National Science Foundation under DMS-2106556, and in part by the Susan M. Smith Professorship.
Acknowledgements
The authors thank Daniel Bartl and Benoît Kloeckner for helpful comments. We are further grateful to an anonymous reviewer and the associate editor for their thorough reading and careful feedback which greatly improved the paper.
Citation
Erhan Bayraktar. Stephan Eckstein. Xin Zhang. "Stability and sample complexity of divergence regularized optimal transport." Bernoulli 31 (1) 213 - 239, February 2025. https://doi.org/10.3150/24-BEJ1725
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