Paper:
Distributionally Robust Optimization (DRO) has been proposed as an alternative to Empirical Risk Minimization (ERM) in order to account for potential biases in the training data distribution. However, its use in deep learning has been severely restricted due to the relative inefficiency of the optimizers available for DRO in comparison to the wide-spread Stochastic Gradient Descent (SGD) based optimizers for deep learning with ERM. In this work, we propose SGD with hardness weighted sampling, a principled and efficient optimization method for DRO in machine learning that is particularly suited in the context of deep learning. Similar to an online hard example mining strategy in essence and in practice, the proposed algorithm is straightforward to implement and computationally as efficient as SGD-based optimizers used for deep learning. It only requires adding a softmax layer and maintaining a history of the loss values for each training example to compute adaptive sampling probabilities. In contrast to typical ad hoc hard mining approaches, and exploiting recent theoretical results in deep learning optimization, we prove the convergence of our DRO algorithm for over-parameterized deep learning networks with ReLU activation and finite number of layers and parameters. Preliminary results demonstrate the feasibility and usefulness of our approach.

Review:
This paper proposes a method for Distributionally Robust Optimization (DRO). DRO has recently been proposed (Namkoong and Duchi 2016 and others) as a robust learning framework compared to Empirical Risk Minimization (ERM).  My analysis of this work in short is that the problem that this paper addresses is interesting and not yet solved. This paper proposes a simple and efficient method for DRO. However, convergence guarantees are weak which is reflected in their weak experimental results. And for a 10-page paper, the writing has to improve quite a lot.
Summary of contributions:
The phi-divergence variation of DRO introduces a min-max objective that minimizes a maximally reweighted empirical risk. The main contribution of this work is as follows:
- Show that for two common phi-divergences (KL-divergence and Pearson-chi2 divergence), the inner maximization has a simple solution for the weights that depends on the value of the loss on the training set.
- Use the above algorithm and modify SGD by changing the sampling of data according to the weights.
- Convergence proofs for the above algorithm for wide networks (not necessarily infinite-width).
Cons:
- Fig 1, ERM should not be so bad as if predicting randomly. This suggests a problem in the experimental setting. At least more ablation study is needed, e.g. try varying the percentage of data kept and show the final test accuracy as a function of this percentage for ERM. In the worst case, the accuracy for ERM should be ~90% on 9 classes and zero on 10th which is ~80% for uniform test accuracy. Another point, what is the train accuracy on cats? ERM should be overfitting to those few cats in the training set and achieve some non-zero accuracy at test. But it seems the test accuracy is almost exactly 0 at test time.
- In Theorem 5.1 that provides convergence proofs, the learning rate eta_exact has a dependence on 1/beta, which means much smaller learning rates should be used for the proposed method. However, in the experiments it seems that the same learning rate is used for all methods. This seems to trouble the convergence very clearly as the curves start to fluctuate considerably by increasing for larger betas. There is no bounds on beta which in practice can force us to use orders of magnitude smaller learning rates.
- Section 4.3 aims at linking hard negative mining to the proposed method. What they actually do is propose a new definition for hard-negative mining which is satisfied by the proposed method. There is little basis for suggesting this definition. There are myriads of work on hard-negative mining and suggesting a new definition needs a more thorough study of related works.
- Section A.1.1 argues we would stop ERM when it plateaus, but it doesn't look like it has plateaued in fig 2 left.
- Section A.3, I'm not sure lr=1 would be stable wide resnet. Maybe there is a problem with the experimental setting.
- There is such much non-crucial details in the main body that increased the main text to 10 pages. At least 2 pages could be saved by moving details of theorems and convergence results to the appendix.
After rebuttal:
I'm keeping more score. The manuscript has been improved quite a lot.  My main concern remains the experiments. Both texts and mathematical statements still need more edits.
The authors have completely removed experiments on cifar10. Paper now has only one set of experiments on MNIST with little ablation study. I still think my suggested ablation study is interesting. As a minimal sanity check, I'd like to see the performance of DRO on the original MNIST dataset. Basically, how much do we lose by using a robust method on a balanced dataset? Even with full ablation studies, I don't think MNIST is not enough for testing methods based on hard-negative mining.