Review for NeurIPS paper: Large-Scale Methods for Distributionally Robust Optimization

Summary and Contributions: The paper studies the use of batch stochastic gradient methods to solve large scale DRO problems. In these scenarios, we face two problems: (1) Stochastic gradient estimates of DRO problems are biased; (2) Due to the size of large-scale problems, the convergence rate of the methods used to tackle them should not depend on either the number of parameters d or number of training examples N. The authors tackled problem (1) by defining a surrogate objective for which the gradient estimates are unbiased. Then, by carefully bounding the difference between the true and surrogate objectives as a function of the batch size n, the authors are able to give optimality bounds for the true cost by optimizing the surrogate cost, using a large enough batch size. Moreover, for some classes of robust risks, the authors also bound the variance of the gradient estimates. This allows them to use an accelerated version of the stochastic gradient method which achieves tighter convergence bounds.