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We consider distributionally robust optimization (DRO) problems, reformulatedas distributionally robust feasibility (DRF) problems, with multipleexpectation constraints. We propose a generic stochastic first-ordermeta-algorithm, where the decision variables and uncertain distributionparameters are each updated separately by applying stochastic first-ordermethods. We then specialize our results to the case of using two specificversions of stochastic mirror descent (SMD): (i) a novel approximate version ofSMD to update the decision variables, and (ii) the bandit mirror descent methodto update the distribution parameters in the case of $\chi^2$-divergence sets.For this specialization, we demonstrate that the total number of iterations isindependent of the dimensions of the decision variables and distributionparameters. Moreover, the cost per iteration to update both sets of variablesis nearly independent of the dimension of the distribution parameters, allowingfor high dimensional ambiguity sets. Furthermore, we show that the total numberof iterations of our algorithm has a logarithmic dependence on the number ofconstraints. Experiments on logistic regression with fairness constraints,personalized parameter selection in a social network, and the multi-itemnewsvendor problem verify the theoretical results and show the usefulness ofthe algorithm, in particular when the dimension of the distribution parametersis large.

Stochastic First-Order Algorithms for Constrained Distributionally Robust Optimization