Outlier Robust Mean Estimation with Subgaussian Rates via Stability

Part of Advances in Neural Information Processing Systems 33 (NeurIPS 2020)

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Ilias Diakonikolas, Daniel M. Kane, Ankit Pensia


We study the problem of outlier robust high-dimensional mean estimation under a bounded covariance assumption, and more broadly under bounded low-degree moment assumptions. We consider a standard stability condition from the recent robust statistics literature and prove that, except with exponentially small failure probability, there exists a large fraction of the inliers satisfying this condition. As a corollary, it follows that a number of recently developed algorithms for robust mean estimation, including iterative filtering and non-convex gradient descent, give optimal error estimators with (near-)subgaussian rates. Previous analyses of these algorithms gave significantly suboptimal rates. As a corollary of our approach, we obtain the first computationally efficient algorithm for outlier robust mean estimation with subgaussian rates under a bounded covariance assumption.