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 $\rho$ (local corruptions). Given access to $n$ such corrupted samples, we seek a computationally efficient estimator $\hat{P}n$ that minimizes the Wasserstein distance $\mathsf{W}1(\hat{P}n,P)$. In fact, we attack the fine-grained task of minimizing $\mathsf{W}1(\Pi# \hat{P}n, \Pi_# P)$ for all orthogonal projections $\Pi \in \mathbb{R}^{d \times d}$, with performance scaling with $\mathrm{rank}(\Pi) = k$. 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 $\sqrt{\varepsilon k} + \rho + d^{{O(1)}\tilde{O}(n{-1/k})$} when $P$ has bounded moments of order $2+\delta$, for constant $\delta > 0$. For data distributions with bounded covariance, our finite-sample bounds match the minimax population-level optimum for large sample sizes. 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.