Robust Stochastic Optimization: Learning the Tails

We develop and analyze a robust stochastic optimization framework that learns a solution which is robust to perturbations in the underlying distribution. We formulate a convex procedure for the finite sample approximation and provide statistical guarantees, showing that the finite sample problem concentrates around the robustified population objective. The robust solutions optimize performance on the tails of the input distribution instead of the average performance. Simulation experiments show that robust solutions outperform the empirical risk minimizer under adversarial perturbations in the underlying distribution by optimizing the performance on the tails of the input distribution.