Weakness: Due to the intractbility of the MMD DRO problem, the submission did not find an exact reformulation as much other literature in DRO did for other probability metrics. Instead, the author provides several layers of approximation. The reason why I emphasize the importance of a tight bound, if not an exact reformulation, is that one of the major criticism about (distributionally) robust optimization is that it is sometimes too conservative, and thus a loose upper bound might not be sufficient to mitigate the over-conservativeness and demonstrate the power of distributionally robust optimization. When a new distance is introduced into the DRO framework, a natural question is why it should be used compared with other existing approaches. I hope there will be a more fair comparision in the camera-ready version. =============== 1. The study of MMD DRO is mostly motivated by the poor out-of-sample performance of existing phi-divergence and Wasserstein uncertainty sets. However, I don't believe this is indeed the case. For example, Namkoong and Duchi (2016), and Blanchet, Kang, and Murthy (2016) show the dimension-independent bound 1/\sqrt{n} for a broad class of objective functions in the case of phi-divergence and Wasserstein metric respectively. They didn't require the population distribution to be within the uncertainty set, and in fact, such a requirement is way too conservative and it is exactly what they wanted to avoid. 2. Unlike phi-divergence or Wasserstein uncertainty sets, MMD DRO seems not enjoy a tractable exact equivalent reformulation, which seems to be a severe drawback to me. The upper bound provided in Theorem 3.1 is crude especially because it drops the nonnegative constraint on the distribution, and further approximation is still needed even applied to a simple kernel ridge regression problem. Moreover, it seems restrictive to assume the loss \ell_f belongs to the RKHS as already pointed out by the authors. 3. I am confused about the statement in Theorem 5.1, as it might indicate some disadvantage of MMD DRO, as it provides a more conservative upper bound than the variance regularized problem. 4. Given the intractability of the MMD DRO and several layers of approximation, the numerical experiment in Section 6 is insufficient to demonstrate the usefulness of the new framework. References: Namkoong, H. and Duchi, J.C., 2017. Variance-based regularization with convex objectives. In Advances in Neural Information Processing Systems (pp. 2971-2980). Blanchet, J., Kang, Y. and Murthy, K., 2016. Robust wasserstein profile inference and applications to machine learning. arXiv preprint arXiv:1610.05627.
3. I am confused about the statement in Theorem 5.1, as it might indicate some disadvantage of MMD DRO, as it provides a more conservative upper bound than the variance regularized problem.