Optimal Robust Estimation under Local and Global Corruptions: Stronger Adversary and Smaller Error

Algorithmic robust statistics has traditionally focused on the contamination model where a small fraction of the samples are arbitrarily corrupted. We consider a recent contamination model that combines two kinds of corruptions: (i) small fraction of arbitrary outliers, as in classical robust statistics, and (ii) local perturbations, where samples may undergo bounded shifts on average. While each noise model is well understood individually, the combined contamination model poses new algorithmic challenges, with only partial results known. Existing efficient algorithms are limited in two ways: (i) they work only for a weak notion of local perturbations, and (ii) they obtain suboptimal error for isotropic subgaussian distributions (among others). The latter limitation led [NGS24, COLT'24] to hypothesize that improving the error might, in fact, be computationally hard. Perhaps surprisingly, we show that information theoretically optimal error can indeed be achieved in polynomial time, under an even \emph{stronger} local perturbation model (the sliced-Wasserstein metric as opposed to the Wasserstein metric). Notably, our analysis reveals that the entire family of stability-based robust mean estimators continues to work optimally in a black-box manner for the combined contamination model. This generalization is particularly useful in real-world scenarios where the specific form of data corruption is not known in advance. We also present efficient algorithms for distribution learning and principal component analysis in the combined contamination model.