With probability at least $1-3\eta$, it holds that

$\displaystyle\left\|\theta(\mathbb{P}_{\star})-\hat{\theta}(\hat{\mathbb{P}}_{n})\right\|_{2}\leq
C\sigma\left(\epsilon\sqrt{\log(\epsilon)}+\sqrt{\frac{d+\log(1/\eta)}{n}}\right),$

where $C$ is a universal constant. The dependence on $\epsilon,d$ and $n$ are
information-theoretically optimal (cf. equation (3.4)).

We remark that minimax optimality of the min-min formulation (3.8) in a wider
context (e.g., Wasserstein contamination) is a prospective area of interest
for future research.

##  4 Conclusions and Discussion

The goal of this section is to briefly discuss various areas of potential
research interest in connection with DRO and statistics. The discussion is not
exhaustive, but rather we want to expose the fact that DRO as a statistical
tool offers a wide range of opportunities for the statistical community.

We mentioned in Section 2.2.2, simply in terms of asymptotic mean squared
error as the sample size increases, it is sensible to ponder on the benefits
of DRO estimators. The work of Lam, (2021) shows that in the presence of
sufficient regularity (e.g. smoothness of the loss) DRO estimators tend to
dominate in second-order stochastic dominance of the empirical risk
minimization estimator of the optimal loss. This observation is consistent
with our discussion in Section 2.2.2. Nevertheless, the situation may be
different if these regularity conditions are not satisfied. For instance,
Duchi and Namkoong, (2018, Section 3.3) shows that in settings involving non-
smooth losses, DRO estimators may enjoy superior rates compared to their
empirical risk minimization counterparts.

Continuing in the context of classical statistical analysis involving large
sample properties. There are objects that the DRO estimation approach offers
that are interesting statistically speaking. The most natural such object is
the worst-case distribution, which is a by-product of the DRO approach and
possesses rich interpretations. Even in the context of the estimators that DRO
recovers exactly and that are well-known in statistics, the DRO approach
furnishes additional insight to these classical estimators using the
associated worst-case distribution.

Another example of an interesting statistical object to study offered by DRO
formulations is the natural confidence region induced by the DRO and discussed
in Subsection 2.2.5. Using the duality between confidence regions and
hypothesis testing we can compare the efficiency of various confidence regions
implied by standard notions of efficiency in hypothesis testing.

Statistical efficiency is also of interest in connection to important
parameters, such as the dimension, for example. We have seen that a suitably
chosen distributional uncertainty region can be used to show the equivalence
between a DRO estimator and a well-known estimator. An example of this
situation is square-root LASSO and the DRO-motivated choice of uncertainty
size recovers regularization prescriptions studied in the high-dimensional
statistics literature. Likewise, in the context of $\phi$-divergence, the DRO-
based estimator is used to re-weight samples in order to hedge against
significant inference errors in subpopulations. In summary, while the DRO
estimator may have a higher asymptotic mean squared error compared to the
empirical risk minimization estimator when used in situations in which the
statistical problem is ill-posed (i.e. the sample size is relatively small
compared to the information required to estimate the parameter) or when the
goal is not purely based on mean-squared error but we are interested in
hedging a different type of risk, then DRO based estimation offers enough
flexibility and interpretability, not only through their formulation but also
through the associated worst-case distribution.

In general, we also note that adding constraints or exploring other types of
distributional uncertainty sets that can be used to better inform the
attributes of the adversary to reduce conservativeness is a significant topic
of research interest. For example, the work of Olea et al., (2022) explores
different DRO uncertainty sets based on the sliced Wasserstein distance. The
advantage of this formulation is that it does not suffer from the statistical
course of dimensionality for comparing distributions in high dimensions (as is
the case of the Wasserstein distance); see also the approaches recently
advocated by Bennouna and Van Parys, (2022); Liu et al., (2023).

Another area of significant interest which we touched only superficially is
the issue of fairness. We mentioned that $\phi$-divergence has been utilized
to try to improve the inference quality in estimated statistics involving
minority sub-populations. Other DRO-based ideas have been recently applied in
the context of fairness. For example, Taskesen et al., (2020); Si et al.,
(2021) propose a projection-based hypothesis test closely related to the one
discussed in Section 2.2.5 for algorithmic fairness. This is a setting in
which the associated distribution induced by DRO-type mechanisms deserves
significantly more statistical investigation.

Next, we comment on dynamic DRO settings. This is an area that closely
connects with what is known as distributionally robust reinforcement learning
and it is in its infancy (see, e.g., Xu and Mannor, (2010); Osogami, (2012);
Lim et al., (2013); Zhou et al., (2021); Backhoff et al., (2022); Si et al.,
(2023)). Even fundamental problems involving how to formulate associated
distributionally robust Markov decision processes based on the sequentially
available information for the agent and the adversary are significantly non-
trivial (see Wang et al., (2023)). This area opens up a wide range of
interesting questions for the statistics community. To give a sense of why DRO
naturally offers a meaningful approach to estimation and optimization in these
settings, note that in many situations of interest in stochastic control,
there is a real possibility of facing unobserved (i.e. confounding) variables.
This type of formulation is naturally posed as a so-called Partially Observed
Markov Decision Process, which is challenging to study since it requires a
history-dependent specification at every point in time. In these settings, the
statistician can introduce a Markovian model (thus reducing the problem to a
standard reinforcement learning environment) and instead use DRO to hedge
against the model misspecification which has been introduced for tractability.

Finally, we finish our discussion by noting that the robust statistics
perspective offered in this paper provides a useful point of view to connect
and contrast DRO estimators and classical robust estimators. This perspective,
characterized by the order in which the statistician and the adversary make
their decision, was introduced in this paper primarily to motivate the
fundamental differences in the nature of these types of robust estimators, The
DRO estimator is pessimistic in nature because the statistician is at the
mercy of an adversary that will change the out-of-sample environment. In
robust statistics, hidden in the data lies useful information about the actual
out-of-sample distribution - the adversary has made its move. Therefore, the
statistician naturally could try to clean or rectify the contamination
employed by the adversary thus leading to an optimistic approach.

## Acknowledgments

The material in this paper is based upon work supported by the Air Force
Office of Scientific Research under award number FA9550-20-1-0397. Additional
support is gratefully acknowledged from NSF 1915967, 2118199, 2229012,
2312204.

## References

  * An and Gao, (2021) An, Y. and Gao, R. (2021).  Generalization bounds for (Wasserstein) robust optimization.  In Advances in Neural Information Processing Systems, volume 34, pages 10382–10392.
  * Aravkin and Davis, (2020) Aravkin, A. and Davis, D. (2020).  Trimmed statistical estimation via variance reduction.  Mathematics of Operations Research, 45(1):292–322.
  * Audibert and Catoni, (2011) Audibert, J.-Y. and Catoni, O. (2011).  Robust linear least squares regression.  Annals of Statistics, 39(5):2766 – 2794.
  * Azizian et al., (2023) Azizian, W., Iutzeler, F., and Malick, J. (2023).  Regularization for wasserstein distributionally robust optimization.  ESAIM: Control, Optimisation and Calculus of Variations, 29:33.
  * Backhoff et al., (2022) Backhoff, J., Bartl, D., Beiglböck, M., and Wiesel, J. (2022).  Estimating processes in adapted Wasserstein distance.  Annals of Applied Probability, 32(1):529–550.
  * Bartl et al., (2021) Bartl, D., Drapeau, S., Obłój, J., and Wiesel, J. (2021).  Sensitivity analysis of Wasserstein distributionally robust optimization problems.  Proceedings of the Royal Society A, 477(2256):20210176.
  * Bateni and Dalalyan, (2019) Bateni, A.-H. and Dalalyan, A. S. (2019).  Confidence regions and minimax rates in outlier-robust estimation on the probability simplex.  Electronic Journal of Statistics.
  * Bauschke and Combettes, (2011) Bauschke, H. and Combettes, P. (2011).  Convex Analysis and Monotone Operator Theory in Hilbert Spaces.  CMS Books in Mathematics. Springer New York.
  * Belloni et al., (2011) Belloni, A., Chernozhukov, V., and Wang, L. (2011).  Square-root lasso: pivotal recovery of sparse signals via conic programming.  Biometrika, 98(4):791–806.
  * Bennett et al., (2023) Bennett, A., Kallus, N., Mao, X., Newey, W., Syrgkanis, V., and Uehara, M. (2023).  Minimax instrumental variable regression and $L_{2}$ convergence guarantees without identification or closedness.  arXiv preprint arXiv:2302.05404.
  * Bennouna and Van Parys, (2022) Bennouna, A. and Van Parys, B. (2022).  Holistic robust data-driven decisions.  arXiv preprint arXiv:2207.09560.
  * Berlinet and Thomas-Agnan, (2011) Berlinet, A. and Thomas-Agnan, C. (2011).  Reproducing Kernel Hilbert Spaces in Probability and Statistics.  Springer Science & Business Media.
  * Bernholt, (2006) Bernholt, T. (2006).  Robust estimators are hard to compute.  Technical Report 2005,52, Universität Dortmund.
  * Bertsimas et al., (2022) Bertsimas, D., Imai, K., and Li, M. L. (2022).  Distributionally robust causal inference with observational data.  arXiv preprint arXiv:2210.08326.
  * Bhatia et al., (2017) Bhatia, K., Jain, P., Kamalaruban, P., and Kar, P. (2017).  Consistent robust regression.  In Advances in Neural Information Processing Systems, volume 30.
  * Bhatia et al., (2015) Bhatia, K., Jain, P., and Kar, P. (2015).  Robust regression via hard thresholding.  In Advances in Neural Information Processing Systems, volume 28.
  * Blanchet and Kang, (2017) Blanchet, J. and Kang, Y. (2017).  Distributionally robust groupwise regularization estimator.  In Asian Conference on Machine Learning, pages 97–112. PMLR.
  * Blanchet and Kang, (2020) Blanchet, J. and Kang, Y. (2020).  Semi-supervised learning based on distributionally robust optimization.  Data Analysis and Applications 3: Computational, Classification, Financial, Statistical and Stochastic Methods, 5:1–33.
  * Blanchet and Kang, (2021) Blanchet, J. and Kang, Y. (2021).  Sample out-of-sample inference based on Wasserstein distance.  Operations Research, 69(3):985–1013.
  * (20) Blanchet, J., Kang, Y., and Murthy, K. (2019a).  Robust Wasserstein profile inference and applications to machine learning.  Journal of Applied Probability, 56(3):830–857.
  * (21) Blanchet, J., Kang, Y., Olea, J. L. M., Nguyen, V. A., and Zhang, X. (2023a).  Dropout training is distributionally robust optimal.  Journal of Machine Learning Research, 24(180):1–60.
  * (22) Blanchet, J., Kang, Y., Zhang, F., He, F., and Hu, Z. (2021a).  Doubly robust data-driven distributionally robust optimization.  Applied Modeling Techniques and Data Analysis 1: Computational Data Analysis Methods and Tools, 7:75–90.
  * (23) Blanchet, J., Kuhn, D., Li, J., and Taskesen, B. (2023b).  Unifying distributionally robust optimization via optimal transport theory.  arXiv preprint arXiv:2308.05414.
  * (24) Blanchet, J., Murthy, K., and Nguyen, V. A. (2021b).  Statistical analysis of wasserstein distributionally robust estimators.  In Tutorials in Operations Research: Emerging Optimization Methods and Modeling Techniques with Applications, pages 227–254. INFORMS.
  * (25) Blanchet, J., Murthy, K., and Si, N. (2022a).  Confidence regions in Wasserstein distributionally robust estimation.  Biometrika, 109(2):295–315.
  * (26) Blanchet, J., Murthy, K., and Zhang, F. (2022b).  Optimal transport-based distributionally robust optimization: Structural properties and iterative schemes.  Mathematics of Operations Research, 47(2):1500–1529.
  * Blanchet and Shapiro, (2023) Blanchet, J. and Shapiro, A. (2023).  Statistical limit theorems in distributionally robust optimization.  arXiv preprint arXiv:2303.14867.
  * (28) Blanchet, J., Zhang, F., Kang, Y., and Hu, Z. (2019b).  A distributionally robust boosting algorithm.  In 2019 Winter Simulation Conference, pages 3728–3739. IEEE.
  * Box, (1976) Box, G. E. (1976).  Science and statistics.  Journal of the American Statistical Association, 71(356):791–799.
  * Box, (1979) Box, G. E. (1979).  Robustness in the strategy of scientific model building.  In Robustness in Statistics, pages 201–236. Elsevier.
  * Box, (1953) Box, G. E. P. (1953).  Non-normality and tests on variances.  Biometrika, 40(3/4):318–335.
  * Chao et al., (2019) Chao, G., Yuan, Y., and Weizhi, Z. (2019).  Robust estimation via generative adversarial networks.  In International Conference on Learning Representations.
  * Charikar et al., (2017) Charikar, M., Steinhardt, J., and Valiant, G. (2017).  Learning from untrusted data.  In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, pages 47–60.
  * Chen et al., (2018) Chen, M., Gao, C., and Ren, Z. (2018).  Robust covariance and scatter matrix estimation under Huber’s contamination model.  Annals of Statistics, 46(5):1932 – 1960.
  * Csiszár, (1975) Csiszár, I. (1975).  I-divergence geometry of probability distributions and minimization problems.  Annals of Probability, pages 146–158.
  * Dapogny et al., (2023) Dapogny, C., Iutzeler, F., Meda, A., and Thibert, B. (2023).  Entropy-regularized wasserstein distributionally robust shape and topology optimization.  Structural and Multidisciplinary Optimization, 66(3):42.
  * Delage and Ye, (2010) Delage, E. and Ye, Y. (2010).  Distributionally robust optimization under moment uncertainty with application to data-driven problems.  Operations Research, 58(3):595–612.
  * Depersin and Lecué, (2022) Depersin, J. and Lecué, G. (2022).  Robust sub-Gaussian estimation of a mean vector in nearly linear time.  Annals of Statistics, 50(1):511 – 536.
  * Devroye et al., (2016) Devroye, L., Lerasle, M., Lugosi, G., and Oliveira, R. I. (2016).  Sub-Gaussian mean estimators.  Annals of Statistics, 44(6):2695 – 2725.
  * (40) Diakonikolas, I., Kamath, G., Kane, D., Li, J., Moitra, A., and Stewart, A. (2019a).  Robust estimators in high-dimensions without the computational intractability.  SIAM Journal on Computing, 48(2):742–864.
  * (41) Diakonikolas, I., Kamath, G., Kane, D., Li, J., Steinhardt, J., and Stewart, A. (2019b).  Sever: A robust meta-algorithm for stochastic optimization.  In International Conference on Machine Learning, pages 1596–1606. PMLR.
  * (42) Diakonikolas, I., Kamath, G., Kane, D. M., Li, J., Moitra, A., and Stewart, A. (2017a).  Being robust (in high dimensions) can be practical.  In International Conference on Machine Learning, pages 999–1008. PMLR.
  * Diakonikolas and Kane, (2023) Diakonikolas, I. and Kane, D. (2023).  Algorithmic High-Dimensional Robust Statistics.  Cambridge University Press.
  * Diakonikolas and Kane, (2019) Diakonikolas, I. and Kane, D. M. (2019).  Recent advances in algorithmic high-dimensional robust statistics.  arXiv preprint arXiv:1911.05911.
  * (45) Diakonikolas, I., Kane, D. M., and Stewart, A. (2017b).  Statistical query lower bounds for robust estimation of high-dimensional Gaussians and Gaussian mixtures.  In 2017 IEEE 58th Annual Symposium on Foundations of Computer Science, pages 73–84.
  * (46) Diakonikolas, I., Kane, D. M., and Stewart, A. (2018a).  Learning geometric concepts with nasty noise.  In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, pages 1061–1073.
  * (47) Diakonikolas, I., Kane, D. M., and Stewart, A. (2018b).  List-decodable robust mean estimation and learning mixtures of spherical gaussians.  In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, pages 1047–1060.
  * Donoho and Montanari, (2016) Donoho, D. and Montanari, A. (2016).  High dimensional robust M-estimation: Asymptotic variance via approximate message passing.  Probability Theory and Related Fields, 166(3):935–969.
  * Donoho, (1994) Donoho, D. L. (1994).  Statistical Estimation and Optimal Recovery.  Annals of Statistics, 22(1):238 – 270.
  * Donoho and Gasko, (1992) Donoho, D. L. and Gasko, M. (1992).  Breakdown properties of location estimates based on halfspace depth and projected outlyingness.  Annals of Statistics, 20(4):1803–1827.
  * Donoho and Huber, (1983) Donoho, D. L. and Huber, P. J. (1983).  The notion of breakdown point.  A festschrift for Erich L. Lehmann, 157184.
  * Donoho and Liu, (1988) Donoho, D. L. and Liu, R. C. (1988).  The “automatic” robustness of minimum distance functionals.  Annals of Statistics, 16(2):552–586.
  * Donoho and Liu, (1991) Donoho, D. L. and Liu, R. C. (1991).  Geometrizing rates of convergence, III.  Annals of Statistics, pages 668–701.
  * Duchi et al., (2023) Duchi, J., Hashimoto, T., and Namkoong, H. (2023).  Distributionally robust losses for latent covariate mixtures.  Operations Research, 71(2):649–664.
  * Duchi and Namkoong, (2018) Duchi, J. and Namkoong, H. (2018).  Variance-based regularization with convex objectives.  Journal of Machine Learning Research, 19:1–55.
  * Duchi et al., (2021) Duchi, J. C., Glynn, P. W., and Namkoong, H. (2021).  Statistics of robust optimization: A generalized empirical likelihood approach.  Mathematics of Operations Research, 46(3):946–969.
  * Duchi and Namkoong, (2021) Duchi, J. C. and Namkoong, H. (2021).  Learning models with uniform performance via distributionally robust optimization.  Annals of Statistics, 49(3):1378–1406.
  * Fournier and Guillin, (2015) Fournier, N. and Guillin, A. (2015).  On the rate of convergence in Wasserstein distance of the empirical measure.  Probability Theory and Related Fields, 162(3-4):707–738.
  * Freund and Schapire, (1997) Freund, Y. and Schapire, R. E. (1997).  A decision-theoretic generalization of on-line learning and an application to boosting.  Journal of Computer and System Sciences, 55(1):119–139.
  * Gao, (2020) Gao, C. (2020).  Robust regression via mutivariate regression depth.  Bernoulli, 26(2):1139 – 1170.
  * Gao, (2022) Gao, R. (2022).  Finite-sample guarantees for Wasserstein distributionally robust optimization: Breaking the curse of dimensionality.  Operations Research.
  * Gao et al., (2022) Gao, R., Chen, X., and Kleywegt, A. J. (2022).  Wasserstein distributionally robust optimization and variation regularization.  Operations Research.
  * Gao and Kleywegt, (2023) Gao, R. and Kleywegt, A. (2023).  Distributionally robust stochastic optimization with Wasserstein distance.  Mathematics of Operations Research, 48(2):603–655.
  * Gao et al., (2018) Gao, R., Xie, L., Xie, Y., and Xu, H. (2018).  Robust hypothesis testing using Wasserstein uncertainty sets.  In Advances in Neural Information Processing Systems, volume 31.
  * Goodfellow et al., (2014) Goodfellow, I., Pouget-Abadie, J., Mirza, M., Xu, B., Warde-Farley, D., Ozair, S., Courville, A., and Bengio, Y. (2014).  Generative adversarial nets.  In Advances in Neural Information Processing Systems, volume 27.
  * Goodfellow et al., (2015) Goodfellow, I. J., Shlens, J., and Szegedy, C. (2015).  Explaining and harnessing adversarial examples.  In International Conference on Learning Representations.
  * Gotoh et al., (2023) Gotoh, J.-y., Kim, M. J., and Lim, A. E. (2023).  A data-driven approach to beating SAA out of sample.  Operations Research.
  * Gül and Zoubir, (2017) Gül, G. and Zoubir, A. M. (2017).  Minimax robust hypothesis testing.  IEEE Transactions on Information Theory, 63(9):5572–5587.
  * Hampel, (1968) Hampel, F. (1968).  Contributions to the Theory of Robust Estimation.  University of California.
  * Hampel, (1971) Hampel, F. R. (1971).  A general qualitative definition of robustness.  Annals of Mathematical Statistics, 42(6):1887 – 1896.
  * He and Lam, (2021) He, S. and Lam, H. (2021).  Higher-order expansion and bartlett correctability of distributionally robust optimization.  arXiv preprint arXiv:2108.05908.
  * Hopkins and Li, (2018) Hopkins, S. B. and Li, J. (2018).  Mixture models, robustness, and sum of squares proofs.  In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, pages 1021–1034.
  * Hu et al., (2018) Hu, W., Niu, G., Sato, I., and Sugiyama, M. (2018).  Does distributionally robust supervised learning give robust classifiers?  In International Conference on Machine Learning, pages 2029–2037. PMLR.
  * Hu and Hong, (2013) Hu, Z. and Hong, L. J. (2013).  Kullback-leibler divergence constrained distributionally robust optimization.  Available at Optimization Online, 1(2):9.
  * Huber, (2004) Huber, P. (2004).  Robust Statistics.  Wiley Series in Probability and Statistics - Applied Probability and Statistics Section Series. Wiley.
  * Huber, (1964) Huber, P. J. (1964).  Robust estimation of a location parameter.  Annals of Mathematical Statistics, 35(1):73–101.
  * Huber, (1968) Huber, P. J. (1968).  Robust confidence limits.  Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 10(4):269–278.
  * Huber, (1972) Huber, P. J. (1972).  The 1972 Wald lecture robust statistics: A review.  Annals of Mathematical Statistics, 43(4):1041 – 1067.
  * Iman et al., (2023) Iman, M., Arabnia, H. R., and Rasheed, K. (2023).  A review of deep transfer learning and recent advancements.  Technologies, 11(2):40.
  * Jiang and Xie, (2023) Jiang, N. and Xie, W. (2023).  Distributionally favorable optimization: A framework for data-driven decision-making with endogenous outliers.  Available at Optimization Online.
  * Johnson and Preparata, (1978) Johnson, D. and Preparata, F. (1978).  The densest hemisphere problem.  Theoretical Computer Science, 6(1):93–107.
  * Joly et al., (2017) Joly, E., Lugosi, G., and Oliveira, R. I. (2017).  On the estimation of the mean of a random vector.  Electronic Journal of Statistics, 11(1):440 – 451.
  * Kantorovich and Rubinshtein, (1958) Kantorovich, L. V. and Rubinshtein, S. (1958).  On a space of totally additive functions.  Vestnik of the St. Petersburg University: Mathematics, 13(7):52–59.
  * Karmalkar et al., (2019) Karmalkar, S., Klivans, A., and Kothari, P. (2019).  List-decodable linear regression.  In Advances in Neural Information Processing Systems, volume 32.
  * Klivans et al., (2018) Klivans, A., Kothari, P. K., and Meka, R. (2018).  Efficient algorithms for outlier-robust regression.  In Conference On Learning Theory, pages 1420–1430. PMLR.
  * Kothari and Steinhardt, (2017) Kothari, P. K. and Steinhardt, J. (2017).  Better agnostic clustering via relaxed tensor norms.  arXiv preprint arXiv:1711.07465.
  * Kuhn et al., (2019) Kuhn, D., Esfahani, P. M., Nguyen, V. A., and Shafieezadeh-Abadeh, S. (2019).  Wasserstein distributionally robust optimization: Theory and applications in machine learning.  In Operations Research & Management Science in the Age of Analytics, pages 130–166. INFORMS.
  * Lai et al., (2016) Lai, K. A., Rao, A. B., and Vempala, S. (2016).  Agnostic estimation of mean and covariance.  In 2016 IEEE 57th Annual Symposium on Foundations of Computer Science, pages 665–674. IEEE Computer Society.
  * Lam, (2016) Lam, H. (2016).  Robust sensitivity analysis for stochastic systems.  Mathematics of Operations Research, 41(4):1248–1275.
  * Lam, (2018) Lam, H. (2018).  Sensitivity to serial dependency of input processes: A robust approach.  Management Science, 64(3):1311–1327.
  * Lam, (2021) Lam, H. (2021).  On the impossibility of statistically improving empirical optimization: A second-order stochastic dominance perspective.  arXiv preprint arXiv:2105.13419.
  * Lam and Zhou, (2017) Lam, H. and Zhou, E. (2017).  The empirical likelihood approach to quantifying uncertainty in sample average approximation.  Operations Research Letters, 45(4):301–307.
  * Lee and Raginsky, (2018) Lee, J. and Raginsky, M. (2018).  Minimax statistical learning with Wasserstein distances.  Advances in Neural Information Processing Systems, 31.
  * Levy and Nikoukhah, (2012) Levy, B. C. and Nikoukhah, R. (2012).  Robust state space filtering under incremental model perturbations subject to a relative entropy tolerance.  IEEE Transactions on Automatic Control, 58(3):682–695.
  * Levy et al., (2020) Levy, D., Carmon, Y., Duchi, J. C., and Sidford, A. (2020).  Large-scale methods for distributionally robust optimization.  In Advances in Neural Information Processing Systems, volume 33, pages 8847–8860.
  * Li et al., (2020) Li, J., Chen, C., and So, A. M.-C. (2020).  Fast epigraphical projection-based incremental algorithms for Wasserstein distributionally robust support vector machine.  In Advances in Neural Information Processing Systems, volume 33, pages 4029–4039.
  * Li et al., (2019) Li, J., Huang, S., and So, A. M.-C. (2019).  A first-order algorithmic framework for distributionally robust logistic regression.  In Advances in Neural Information Processing Systems, volume 32.
  * Li et al., (2022) Li, J., Lin, S., Blanchet, J., and Nguyen, V. A. (2022).  Tikhonov regularization is optimal transport robust under martingale constraints.  In Advances in Neural Information Processing Systems, volume 35, pages 17677–17689.
  * Lim et al., (2013) Lim, S. H., Xu, H., and Mannor, S. (2013).  Reinforcement learning in robust Markov decision processes.  In Advances in Neural Information Processing Systems, volume 26.
  * Liu and Gao, (2019) Liu, H. and Gao, C. (2019).  Density estimation with contamination: minimax rates and theory of adaptation.  Electronic Journal of Statistics, 13(2):3613 – 3653.
  * Liu et al., (2020) Liu, L., Shen, Y., Li, T., and Caramanis, C. (2020).  High dimensional robust sparse regression.  In International Conference on Artificial Intelligence and Statistics, pages 411–421. PMLR.
  * Liu and Loh, (2022) Liu, Z. and Loh, P.-L. (2022).  Robust W-GAN-based estimation under Wasserstein contamination.  Information and Inference: A Journal of the IMA, 12(1):312–362.
  * Liu et al., (2023) Liu, Z., Van Parys, B. P., and Lam, H. (2023).  Smoothed $f$-divergence distributionally robust optimization: Exponential rate efficiency and complexity-free calibration.  arXiv preprint arXiv:2306.14041.
  * Lotidis et al., (2023) Lotidis, K., Bambos, N., Blanchet, J., and Li, J. (2023).  Wasserstein distributionally robust linear-quadratic estimation under martingale constraints.  In International Conference on Artificial Intelligence and Statistics, pages 8629–8644. PMLR.
  * Lugosi and Mendelson, (2019) Lugosi, G. and Mendelson, S. (2019).  Sub-Gaussian estimators of the mean of a random vector.  Annals of Statistics, 47(2):783 – 794.
  * Madry et al., (2017) Madry, A., Makelov, A., Schmidt, L., Tsipras, D., and Vladu, A. (2017).  Towards deep learning models resistant to adversarial attacks.  arXiv preprint arXiv:1706.06083.
  * Minsker, (2015) Minsker, S. (2015).  Geometric median and robust estimation in Banach spaces.  Bernoulli, 21(4):2308 – 2335.
  * Mohajerin Esfahani and Kuhn, (2018) Mohajerin Esfahani, P. and Kuhn, D. (2018).  Data-driven distributionally robust optimization using the Wasserstein metric: Performance guarantees and tractable reformulations.  Mathematical Programming, 171(1-2):115–166.
  * Ng, (2004) Ng, A. Y. (2004).  Feature selection, $L_{1}$ vs. $L_{2}$ regularization, and rotational invariance.  In Proceedings of the Twenty-First International Conference on Machine Learning, page 78.
  * Nguyen et al., (2023) Nguyen, V. A., Shafieezadeh-Abadeh, S., Kuhn, D., and Mohajerin Esfahani, P. (2023).  Bridging Bayesian and minimax mean square error estimation via Wasserstein distributionally robust optimization.  Mathematics of Operations Research, 48(1):1–37.
  * Nguyen et al., (2019) Nguyen, V. A., Shafieezadeh Abadeh, S., Yue, M.-C., Kuhn, D., and Wiesemann, W. (2019).  Optimistic distributionally robust optimization for nonparametric likelihood approximation.  In Advances in Neural Information Processing Systems, volume 32.
  * (112) Nguyen, V. A., Si, N., and Blanchet, J. (2020a).  Robust Bayesian classification using an optimistic score ratio.  In International Conference on Machine Learning, pages 7327–7337. PMLR.
  * (113) Nguyen, V. A., Zhang, F., Blanchet, J., Delage, E., and Ye, Y. (2020b).  Distributionally robust local non-parametric conditional estimation.  Advances in Neural Information Processing Systems, 33:15232–15242.
  * Nguyen et al., (2021) Nguyen, V. A., Zhang, F., Blanchet, J., Delage, E., and Ye, Y. (2021).  Robustifying conditional portfolio decisions via optimal transport.  arXiv preprint arXiv:2103.16451.
  * (115) Nguyen, V. A., Zhang, X., Blanchet, J., and Georghiou, A. (2020c).  Distributionally robust parametric maximum likelihood estimation.  In Advances in Neural Information Processing Systems, volume 33, pages 7922–7932.
  * Norton et al., (2017) Norton, M., Takeda, A., and Mafusalov, A. (2017).  Optimistic robust optimization with applications to machine learning.  arXiv preprint arXiv:1711.07511.
  * Olea et al., (2022) Olea, J. L. M., Rush, C., Velez, A., and Wiesel, J. (2022).  On the generalization error of norm penalty linear regression models.  arXiv preprint arXiv:2211.07608.
  * Osogami, (2012) Osogami, T. (2012).  Robustness and risk-sensitivity in Markov decision processes.  In Advances in Neural Information Processing Systems, volume 25.
  * Owen, (2001) Owen, A. B. (2001).  Empirical Likelihood.  CRC press.
  * Peyré et al., (2019) Peyré, G., Cuturi, M., et al. (2019).  Computational optimal transport: With applications to data science.  Foundations and Trends® in Machine Learning, 11(5-6):355–607.
  * Prasad et al., (2020) Prasad, A., Suggala, A. S., Balakrishnan, S., and Ravikumar, P. (2020).  Robust estimation via robust gradient estimation.  Journal of the Royal Statistical Society Series B: Statistical Methodology, 82(3):601–627.
  * Raghavendra and Yau, (2020) Raghavendra, P. and Yau, M. (2020).  List decodable learning via sum of squares.  In Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 161–180. SIAM.
  * Rahimian and Mehrotra, (2022) Rahimian, H. and Mehrotra, S. (2022).  Frameworks and results in distributionally robust optimization.  Open Journal of Mathematical Optimization, 3:1–85.
  * Rockafellar, (1974) Rockafellar, R. (1974).  Conjugate Duality and Optimization.  Society for Industrial and Applied Mathematics.
  * Rockafellar, (1985) Rockafellar, R. (1985).  Extensions of subgradient calculus with applications to optimization.  Nonlinear Analysis: Theory, Methods & Applications, 9(7):665–698.
  * Rockafellar, (1997) Rockafellar, R. (1997).  Convex Analysis.  Princeton Landmarks in Mathematics and Physics. Princeton University Press.
  * Rockafellar, (1963) Rockafellar, R. T. (1963).  Convex Functions and Dual Extremum Problems.  Phd thesis, University of Washington.
  * Rockafellar, (2023) Rockafellar, R. T. (2023).  Distributional robustness, stochastic divergences, and the quadrangle of risk.
  * Rothenhäusler and Bühlmann, (2023) Rothenhäusler, D. and Bühlmann, P. (2023).  Distributionally robust and generalizable inference.  Statistical Science, 38(4):527–542.
  * Royset and Wets, (2022) Royset, J. and Wets, R. (2022).  An Optimization Primer.  Springer Series in Operations Research and Financial Engineering. Springer International Publishing.
  * Royset, (2021) Royset, J. O. (2021).  Good and bad optimization models: Insights from Rockafellians.  In Tutorials in Operations Research: Emerging Optimization Methods and Modeling Techniques with Applications, pages 131–160. INFORMS.
  * Royset et al., (2023) Royset, J. O., Chen, L. L., and Eckstrand, E. (2023).  Rockafellian relaxation in optimization under uncertainty: Asymptotically exact formulations.  arXiv preprint arXiv:2204.04762.
  * Ruszczyński and Shapiro, (2006) Ruszczyński, A. and Shapiro, A. (2006).  Optimization of risk measures.  Probabilistic and Randomized Methods for Design under Uncertainty, pages 119–157.
  * Sagawa et al., (2020) Sagawa, S., Koh, P. W., Hashimoto, T. B., and Liang, P. (2020).  Distributionally robust neural networks.  In International Conference on Learning Representations.
  * Santambrogio, (2015) Santambrogio, F. (2015).  Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling, volume 87.  Birkhäuser.
  * Scarf, (1958) Scarf, H. (1958).  A min-max solution of an inventory problem.  In Studies in the Mathematical Theory of Inventory and Production, pages 201–209. Stanford University Press.
  * Scheffe and Tukey, (1944) Scheffe, H. and Tukey, J. W. (1944).  A formula for sample sizes for population tolerance limits.  Annals of Mathematical Statistics, 15(2):217.
  * Scheffe and Tukey, (1945) Scheffe, H. and Tukey, J. W. (1945).  Non-parametric estimation. I. Validation of order statistics.  Annals of Mathematical Statistics, 16(2):187 – 192.
  * Shafieezadeh-Abadeh et al., (2023) Shafieezadeh-Abadeh, S., Aolaritei, L., Dörfler, F., and Kuhn, D. (2023).  New perspectives on regularization and computation in optimal transport-based distributionally robust optimization.  arXiv preprint arXiv:2303.03900.
  * Shafieezadeh-Abadeh et al., (2019) Shafieezadeh-Abadeh, S., Kuhn, D., and Esfahani, P. M. (2019).  Regularization via mass transportation.  Journal of Machine Learning Research, 20(103):1–68.
  * Shafieezadeh Abadeh et al., (2018) Shafieezadeh Abadeh, S., Nguyen, V. A., Kuhn, D., and Mohajerin Esfahani, P. M. (2018).  Wasserstein distributionally robust Kalman filtering.  In Advances in Neural Information Processing Systems, volume 31.
  * Shapiro, (2017) Shapiro, A. (2017).  Distributionally robust stochastic programming.  SIAM Journal on Optimization, 27(4):2258–2275.
  * Si et al., (2021) Si, N., Murthy, K., Blanchet, J., and Nguyen, V. A. (2021).  Testing group fairness via optimal transport projections.  In International Conference on Machine Learning, pages 9649–9659. PMLR.
  * Si et al., (2023) Si, N., Zhang, F., Zhou, Z., and Blanchet, J. (2023).  Distributionally robust batch contextual bandits.  Management Science.
  * Sinha et al., (2018) Sinha, A., Namkoong, H., and Duchi, J. (2018).  Certifying some distributional robustness with principled adversarial training.  In International Conference on Learning Representations.
  * Staib and Jegelka, (2019) Staib, M. and Jegelka, S. (2019).  Distributionally robust optimization and generalization in kernel methods.  In Advances in Neural Information Processing Systems, volume 32.
  * Steinhardt et al., (2018) Steinhardt, J., Charikar, M., and Valiant, G. (2018).  Resilience: A criterion for learning in the presence of arbitrary outliers.  In 9th Innovations in Theoretical Computer Science Conference, volume 94, pages 45:1–45:21.
  * Steinhardt et al., (2017) Steinhardt, J., Koh, P. W., and Liang, P. (2017).  Certified defenses for data poisoning attacks.  In Advances in Neural Information Processing Systems, volume 30, page 3520–3532.
  * Stone, (1977) Stone, C. J. (1977).  Consistent nonparametric regression.  The annals of statistics, pages 595–620.
  * Strassen, (1965) Strassen, V. (1965).  The existence of probability measures with given marginals.  Annals of Mathematical Statistics, 36(2):423–439.
  * Suggala et al., (2019) Suggala, A. S., Bhatia, K., Ravikumar, P., and Jain, P. (2019).  Adaptive hard thresholding for near-optimal consistent robust regression.  In Conference on Learning Theory, pages 2892–2897. PMLR.
  * Sun and Zou, (2021) Sun, Z. and Zou, S. (2021).  A data-driven approach to robust hypothesis testing using kernel mmd uncertainty sets.  In 2021 IEEE International Symposium on Information Theory (ISIT), pages 3056–3061. IEEE.
  * Székely, (1989) Székely, G. J. (1989).  Potential and kinetic energy in statistics.  Lecture Notes, Budapest Institute of Technology (Technical University).
  * Taskesen et al., (2020) Taskesen, B., Nguyen, V. A., Kuhn, D., and Blanchet, J. (2020).  A distributionally robust approach to fair classification.  arXiv preprint arXiv:2007.09530.
  * Taskesen et al., (2021) Taskesen, B., Yue, M.-C., Blanchet, J., Kuhn, D., and Nguyen, V. A. (2021).  Sequential domain adaptation by synthesizing distributionally robust experts.  In International Conference on Machine Learning, pages 10162–10172. PMLR.
  * Tibshirani, (1996) Tibshirani, R. (1996).  Regression shrinkage and selection via the lasso.  Journal of the Royal Statistical Society Series B: Statistical Methodology, 58(1):267–288.
  * Tukey, (1960) Tukey, J. W. (1960).  A survey of sampling from contaminated distributions.  Contributions to Probability and Statistics: Essays in Honor of Harold Hotelling, pages 448–485.
  * Tukey, (1962) Tukey, J. W. (1962).  The future of data analysis.  Annals of Mathematical Statistics, 33(1):1–67.
  * Tukey, (1975) Tukey, J. W. (1975).  Mathematics and the picturing of data.  In Proceedings of the International Congress of Mathematicians, volume 2, page 523–531.
  * Vaart, (1998) Vaart, A. W. v. d. (1998).  Asymptotic Statistics.  Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press.
  * Van Parys et al., (2021) Van Parys, B. P., Esfahani, P. M., and Kuhn, D. (2021).  From data to decisions: Distributionally robust optimization is optimal.  Management Science, 67(6):3387–3402.
  * Villani et al., (2009) Villani, C. et al. (2009).  Optimal Transport: Old and New, volume 338.  Springer.
  * Wang et al., (2021) Wang, J., Gao, R., and Xie, Y. (2021).  Sinkhorn distributionally robust optimization.  arXiv preprint arXiv:2109.11926.
  * Wang et al., (2023) Wang, S., Si, N., Blanchet, J., and Zhou, Z. (2023).  On the foundation of distributionally robust reinforcement learning.  arXiv preprint arXiv:2311.09018.
  * Watson and Holmes, (2016) Watson, J. and Holmes, C. (2016).  Approximate models and robust decisions.  Statistical Science, 31(4):465–489.
  * Weiss et al., (2016) Weiss, K., Khoshgoftaar, T. M., and Wang, D. (2016).  A survey of transfer learning.  Journal of Big data, 3(1):1–40.
  * Wilson and Cook, (2020) Wilson, G. and Cook, D. J. (2020).  A survey of unsupervised deep domain adaptation.  ACM Transactions on Intelligent Systems and Technology (TIST), 11(5):1–46.
  * Wu et al., (2020) Wu, K., Ding, G. W., Huang, R., and Yu, Y. (2020).  On minimax optimality of gans for robust mean estimation.  In International Conference on Artificial Intelligence and Statistics, volume 108, pages 4541–4551. PMLR.
  * Xu and Mannor, (2010) Xu, H. and Mannor, S. (2010).  Distributionally robust Markov decision processes.  In Advances in Neural Information Processing Systems, volume 23.
  * Zalinescu, (2002) Zalinescu, C. (2002).  Convex Analysis in General Vector Spaces.  G - Reference, Information and Interdisciplinary Subjects Series. World Scientific.
  * (171) Zhang, X., Blanchet, J., Ghosh, S., and Squillante, M. S. (2022a).  A class of geometric structures in transfer learning: Minimax bounds and optimality.  In International Conference on Artificial Intelligence and Statistics, pages 3794–3820. PMLR.
  * (172) Zhang, X., Blanchet, J., Marzouk, Y., Nguyen, V. A., and Wang, S. (2022b).  Distributionally robust Gaussian process regression and Bayesian inverse problems.  arXiv preprint arXiv:2205.13111.
  * Zhou et al., (2021) Zhou, Z., Zhou, Z., Bai, Q., Qiu, L., Blanchet, J., and Glynn, P. (2021).  Finite-sample regret bound for distributionally robust offline tabular reinforcement learning.  In International Conference on Artificial Intelligence and Statistics, pages 3331–3339. PMLR.
  * Zhu et al., (2022) Zhu, B., Jiao, J., and Steinhardt, J. (2022).  Generalized resilience and robust statistics.  Annals of Statistics, 50(4):2256 – 2283.
  * Zhu et al., (2021) Zhu, J.-J., Jitkrittum, W., Diehl, M., and Schölkopf, B. (2021).  Kernel distributionally robust optimization: Generalized duality theorem and stochastic approximation.  In International Conference on Artificial Intelligence and Statistics, pages 280–288. PMLR.
  * Zorzi, (2016) Zorzi, M. (2016).  Robust Kalman filtering under model perturbations.  IEEE Transactions on Automatic Control, 62(6):2902–2907.