  A common goal in statistics and machine learning is to learn models that can
perform well against distributional shifts, such as latent heterogeneous
subpopulations, unknown covariate shifts, or unmodeled temporal effects. We
develop and analyze a distributionally robust stochastic optimization (DRO)
framework that learns a model providing good performance against perturbations
to the data-generating distribution. We give a convex formulation for the
problem, providing several convergence guarantees. We prove finite-sample
minimax upper and lower bounds, showing that distributional robustness
sometimes comes at a cost in convergence rates. We give limit theorems for the
learned parameters, where we fully specify the limiting distribution so that
confidence intervals can be computed. On real tasks including generalizing to
unknown subpopulations, fine-grained recognition, and providing good tail
performance, the distributionally robust approach often exhibits improved
performance.
