Title: SGD with Hardness Weighted Sampling for Distributionally Robust Deep Learning. Abstract: 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 compared to the wide-spread Stochastic Gradient Descent (SGD) based optimizers for deep learning with ERM. In this work, we demonstrate that SGD with hardness weighted sampling is a principled and efficient optimization method for DRO in machine learning and is particularly suited in the context of deep learning. Similar to a 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 an 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.