source_v1/main.bbl

\begin{thebibliography}{}

\bibitem[Ambrosio et~al., 2005]{ambrosio2008gradient}
Ambrosio, L., Gigli, N., and Savare, G. (2005).
\newblock {\em Gradient Flows: {{In}} Metric Spaces and in the Space of Probability Measures}.
\newblock Springer Science \& Business Media.

\bibitem[Ba et~al., 2021]{ba2021understanding}
Ba, J., Erdogdu, M.~A., Ghassemi, M., Sun, S., Suzuki, T., Wu, D., and Zhang, T. (2021).
\newblock Understanding the variance collapse of svgd in high dimensions.
\newblock In {\em International Conference on Learning Representations}.

\bibitem[{Ben-Tal} et~al., 2013]{ben-talRobustSolutionsOptimization2013}
{Ben-Tal}, A., {den Hertog}, D., De~Waegenaere, A., Melenberg, B., and Rennen, G. (2013).
\newblock Robust {{Solutions}} of {{Optimization Problems Affected}} by {{Uncertain Probabilities}}.
\newblock {\em Management Science}, 59(2):341--357.

\bibitem[Blanchet and Glynn, 2015]{blanchet2015unbiased}
Blanchet, J.~H. and Glynn, P.~W. (2015).
\newblock Unbiased monte carlo for optimization and functions of expectations via multi-level randomization.
\newblock In {\em 2015 Winter Simulation Conference (WSC)}, pages 3656--3667. IEEE.

\bibitem[Carrillo et~al., 2024]{carrillo2024fisher}
Carrillo, J.~A., Chen, Y., Huang, D.~Z., Huang, J., and Wei, D. (2024).
\newblock Fisher-rao gradient flow: geodesic convexity and functional inequalities.
\newblock {\em arXiv preprint arXiv:2407.15693}.

\bibitem[Chen et~al., 2022]{chen2022improved}
Chen, Y., Chewi, S., Salim, A., and Wibisono, A. (2022).
\newblock Improved analysis for a proximal algorithm for sampling.
\newblock In {\em Conference on Learning Theory}, pages 2984--3014. PMLR.

\bibitem[Chen et~al., 2023]{chen2023sampling}
Chen, Y., Huang, D.~Z., Huang, J., Reich, S., and Stuart, A.~M. (2023).
\newblock Sampling via gradient flows in the space of probability measures.
\newblock {\em arXiv preprint arXiv:2310.03597}.

\bibitem[Chewi et~al., 2022]{chewi2022query}
Chewi, S., Gerber, P.~R., Lu, C., Le~Gouic, T., and Rigollet, P. (2022).
\newblock The query complexity of sampling from strongly log-concave distributions in one dimension.
\newblock In {\em Conference on Learning Theory}, pages 2041--2059. PMLR.

\bibitem[Chewi et~al., 2024]{chewi2024statistical}
Chewi, S., Niles-Weed, J., and Rigollet, P. (2024).
\newblock Statistical optimal transport.
\newblock {\em arXiv preprint arXiv:2407.18163}.

\bibitem[Conger et~al., 2023]{congerStrategicDistributionShift2023}
Conger, L.~E., Hoffman, F., Mazumdar, E., and Ratliff, L.~J. (2023).
\newblock Strategic {{Distribution Shift}} of {{Interacting Agents}} via {{Coupled Gradient Flows}}.
\newblock In {\em Thirty-Seventh {{Conference}} on {{Neural Information Processing Systems}}}.

\bibitem[Delage and Ye, 2010]{delageDistributionallyRobustOptimization2010}
Delage, E. and Ye, Y. (2010).
\newblock Distributionally robust optimization under moment uncertainty with application to data-driven problems.
\newblock {\em Operations research}, 58(3):595--612.

\bibitem[El~Ghaoui and Lebret, 1997]{el1997robust}
El~Ghaoui, L. and Lebret, H. (1997).
\newblock Robust solutions to least-squares problems with uncertain data.
\newblock {\em SIAM Journal on matrix analysis and applications}, 18(4):1035--1064.

\bibitem[Gao and Kleywegt, 2016]{gaoDistributionallyRobustStochastic2016}
Gao, R. and Kleywegt, A.~J. (2016).
\newblock Distributionally {{Robust Stochastic Optimization}} with {{Wasserstein Distance}}.
\newblock {\em arXiv preprint arXiv:1604.02199}.

\bibitem[Garc{\'\i}a~Trillos and Garc{\'\i}a~Trillos, 2024]{trillosAdversarialRobustnessUse2023}
Garc{\'\i}a~Trillos, C.~A. and Garc{\'\i}a~Trillos, N. (2024).
\newblock On adversarial robustness and the use of wasserstein ascent-descent dynamics to enforce it.
\newblock {\em Information and Inference: A Journal of the IMA}, 13(3):iaae018.

\bibitem[Garc{\'\i}a~Trillos and Sanz-Alonso, 2018]{garcia2018continuum}
Garc{\'\i}a~Trillos, N. and Sanz-Alonso, D. (2018).
\newblock Continuum limits of posteriors in graph bayesian inverse problems.
\newblock {\em SIAM Journal on Mathematical Analysis}, 50(4):4020--4040.

\bibitem[Ghadimi and Lan, 2013]{ghadimi2013stochastic}
Ghadimi, S. and Lan, G. (2013).
\newblock Stochastic first-and zeroth-order methods for nonconvex stochastic programming.
\newblock {\em SIAM journal on optimization}, 23(4):2341--2368.

\bibitem[Gross, 1975]{gross1975logarithmic}
Gross, L. (1975).
\newblock Logarithmic sobolev inequalities.
\newblock {\em American Journal of Mathematics}, 97(4):1061--1083.

\bibitem[Hu and Hong, 2013]{hu2013kullback}
Hu, Z. and Hong, L.~J. (2013).
\newblock Kullback-leibler divergence constrained distributionally robust optimization.
\newblock {\em Available at Optimization Online}, 1(2):9.

\bibitem[Krizhevsky et~al., 2009]{krizhevsky2009learning}
Krizhevsky, A., Hinton, G., et~al. (2009).
\newblock Learning multiple layers of features from tiny images.(2009).

\bibitem[Kuhn et~al., 2025]{kuhnDistributionallyRobustOptimization2024}
Kuhn, D., Shafiee, S., and Wiesemann, W. (2025).
\newblock Distributionally robust optimization.
\newblock {\em Acta Numerica}, 34:579--804.

\bibitem[Lee et~al., 2021]{lee2021structured}
Lee, Y.~T., Shen, R., and Tian, K. (2021).
\newblock Structured logconcave sampling with a restricted gaussian oracle.
\newblock In {\em Conference on Learning Theory}, pages 2993--3050. PMLR.

\bibitem[Levy et~al., 2020]{levyLargeScaleMethodsDistributionally2020}
Levy, D., Carmon, Y., Duchi, J.~C., and Sidford, A. (2020).
\newblock Large-scale methods for distributionally robust optimization.
\newblock {\em Advances in neural information processing systems}, 33:8847--8860.

\bibitem[Liu and Wang, 2016]{liu2016stein}
Liu, Q. and Wang, D. (2016).
\newblock Stein variational gradient descent: A general purpose bayesian inference algorithm.
\newblock {\em Advances in neural information processing systems}, 29.

\bibitem[Lu et~al., 2019]{luAcceleratingLangevinSampling2019}
Lu, Y., Lu, J., and Nolen, J. (2019).
\newblock Accelerating langevin sampling with birth-death.
\newblock {\em arXiv preprint arXiv:1905.09863}.

\bibitem[Lu et~al., 2023]{luBirthdeathDynamicsSampling2023}
Lu, Y., Slep{\v c}ev, D., and Wang, L. (2023).
\newblock Birth-death dynamics for sampling: {{Global}} convergence, approximations and their asymptotics.
\newblock {\em Nonlinearity}, 36(11):5731--5772.

\bibitem[Mielke, 2023]{mielke2023introduction}
Mielke, A. (2023).
\newblock An introduction to the analysis of gradients systems.
\newblock {\em arXiv preprint arXiv:2306.05026}.

\bibitem[Mielke, 2025]{mielkeNotesHellingerDistance2025}
Mielke, A. (2025).
\newblock Some notes on the hellinger distance and various fisher-rao distances.
\newblock {\em arXiv preprint arXiv:2510.02537}.

\bibitem[Mielke and Zhu, 2025]{mielke2025hellinger}
Mielke, A. and Zhu, J.-J. (2025).
\newblock Hellinger-kantorovich gradient flows: Global exponential decay of entropy functionals.
\newblock {\em arXiv preprint arXiv:2501.17049}.

\bibitem[Mohajerin~Esfahani and Kuhn, 2018]{mohajerin2018data}
Mohajerin~Esfahani, P. and Kuhn, D. (2018).
\newblock Data-driven distributionally robust optimization using the wasserstein metric: Performance guarantees and tractable reformulations.
\newblock {\em Mathematical Programming}, 171(1):115--166.

\bibitem[Otto, 1996]{otto1996double}
Otto, F. (1996).
\newblock {\em Double degenerate diffusion equations as steepest descent}.
\newblock Sonderforschungsbereich 256.

\bibitem[Salim et~al., 2022]{salim2022convergence}
Salim, A., Sun, L., and Richtarik, P. (2022).
\newblock A convergence theory for svgd in the population limit under talagrand’s inequality t1.
\newblock In {\em International Conference on Machine Learning}, pages 19139--19152. PMLR.

\bibitem[Shi and Mackey, 2023]{shi2023finite}
Shi, J. and Mackey, L. (2023).
\newblock A finite-particle convergence rate for stein variational gradient descent.
\newblock {\em Advances in Neural Information Processing Systems}, 36:26831--26844.

\bibitem[Sinha et~al., 2017]{sinha2020certifyingdistributionalrobustnessprincipled}
Sinha, A., Namkoong, H., Volpi, R., and Duchi, J. (2017).
\newblock Certifying some distributional robustness with principled adversarial training.
\newblock {\em arXiv preprint arXiv:1710.10571}.

\bibitem[Vempala and Wibisono, 2019]{vempala2019rapid}
Vempala, S. and Wibisono, A. (2019).
\newblock Rapid convergence of the unadjusted langevin algorithm: Isoperimetry suffices.
\newblock {\em Advances in neural information processing systems}, 32.

\bibitem[Wang and Chizat, 2022]{wang2022exponentially}
Wang, G. and Chizat, L. (2022).
\newblock An exponentially converging particle method for the mixed nash equilibrium of continuous games.
\newblock {\em arXiv preprint arXiv:2211.01280}.

\bibitem[Wang et~al., 2021]{wang2021sinkhorn}
Wang, J., Gao, R., and Xie, Y. (2021).
\newblock Sinkhorn distributionally robust optimization.
\newblock {\em arXiv preprint arXiv:2109.11926}.

\bibitem[Wibisono, 2025]{wibisono2025mixing}
Wibisono, A. (2025).
\newblock Mixing time of the proximal sampler in relative fisher information via strong data processing inequality.
\newblock {\em arXiv preprint arXiv:2502.05623}.

\bibitem[Xu et~al., 2024]{xu2024flow}
Xu, C., Lee, J., Cheng, X., and Xie, Y. (2024).
\newblock Flow-based distributionally robust optimization.
\newblock {\em IEEE Journal on Selected Areas in Information Theory}, 5:62--77.

\bibitem[Yu et~al., 2022]{yuFastDistributionallyRobust2022}
Yu, Y., Lin, T., Mazumdar, E.~V., and Jordan, M. (2022).
\newblock Fast {{Distributionally Robust Learning}} with {{Variance-Reduced Min-Max Optimization}}.
\newblock In {\em Proceedings of {{The}} 25th {{International Conference}} on {{Artificial Intelligence}} and {{Statistics}}}, pages 1219--1250. PMLR.

\bibitem[Zhao and Guan, 2018]{zhaoDatadrivenRiskaverseStochastic2018}
Zhao, C. and Guan, Y. (2018).
\newblock Data-driven risk-averse stochastic optimization with {{Wasserstein}} metric.
\newblock {\em Operations Research Letters}, 46(2):262--267.

\bibitem[Zhu et~al., 2021]{zhu2021kernel}
Zhu, J.-J., Jitkrittum, W., Diehl, M., and Sch{\"o}lkopf, B. (2021).
\newblock Kernel distributionally robust optimization: Generalized duality theorem and stochastic approximation.
\newblock In {\em International Conference on Artificial Intelligence and Statistics}, pages 280--288. PMLR.

\bibitem[Zhu and Xie, 2024]{zhu2024distributionally}
Zhu, L. and Xie, Y. (2024).
\newblock Distributionally robust optimization via iterative algorithms in continuous probability spaces.
\newblock {\em arXiv preprint arXiv:2412.20556}.

\end{thebibliography}