Robust Distribution Learning with Local and Global Adversarial Corruptions

We consider learning in an adversarial environment, where an varepsilon-fraction of samples from a distribution P are arbitrarily modified (global corruptions) and the remaining perturbations have average magnitude bounded by ρ (local corruptions). Given access to n such corrupted samples, we seek a computationally efficient estimator hatPn that minimizes the Wasserstein distance mathsfW1(hatPn,P). In fact, we attack the fine-grained task of minimizing mathsfW1(Pi_# hatPn, Pi_# P) for all orthogonal projections Pi ∈ ℝd × d, with performance scaling with rank(Pi) = k. This allows us to account simultaneously for mean estimation (k=1), distribution estimation (k=d), as well as the settings interpolating between these two extremes. We characterize the optimal population-limit risk for this task and then develop an efficient finite-sample algorithm with error bounded by √varepsilon k + ρ + tildeO(d√kn-1/(k lor 2)) when P has bounded covariance. This guarantee holds uniformly in k and is minimax optimal up to the sub-optimality of the plug-in estimator when ρ = varepsilon = 0. Our efficient procedure relies on a novel trace norm approximation of an ideal yet intractable 2-Wasserstein projection estimator. We apply this algorithm to robust stochastic optimization, and, in the process, uncover a new method for overcoming the curse of dimensionality in Wasserstein distributionally robust optimization.