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In this paper, we propose a practical online method for solving a class ofdistributionally robust optimization (DRO) with non-convex objectives, whichhas important applications in machine learning for improving the robustness ofneural networks. In the literature, most methods for solving DRO are based onstochastic primal-dual methods. However, primal-dual methods for DRO sufferfrom several drawbacks: (1) manipulating a high-dimensional dual variablecorresponding to the size of data is time expensive; (2) they are not friendlyto online learning where data is coming sequentially. To address these issues,we consider a class of DRO with an KL divergence regularization on the dualvariables, transform the min-max problem into a compositional minimizationproblem, and propose practical duality-free online stochastic methods withoutrequiring a large mini-batch size. We establish the state-of-the-artcomplexities of the proposed methods with and without a Polyak-\L ojasiewicz(PL) condition of the objective. Empirical studies on large-scale deep learningtasks (i) demonstrate that our method can speed up the training by more than 2times than baseline methods and save days of training time on a large-scaledataset with $\sim$ 265K images, and (ii) verify the supreme performance of DROover Empirical Risk Minimization (ERM) on imbalanced datasets. Of independentinterest, the proposed method can be also used for solving a family ofstochastic compositional problems with state-of-the-art complexities.

An Online Method for A Class of Distributionally Robust Optimization with Non-Convex Objectives