List Decodable Mean Estimation in Nearly Linear Time

Learning from data in the presence of outliers is a fundamental problem in statistics. Until recently, no computationally efficient algorithms were known to compute the mean of a high dimensional distribution under natural assumptions in the presence of even a small fraction of outliers. In this paper, we consider robust statistics in the presence of overwhelming outliers where the majority of the dataset is introduced adversarially. With only an $\alpha < 1/2$ fraction of "inliers" (clean data) the mean of a distribution is unidentifiable. However, in their influential work, [CSV17] introduces a polynomial time algorithm recovering the mean of distributions with bounded covariance by outputting a succinct list of $O(1/\alpha)$ candidate solutions, one of which is guaranteed to be close to the true distributional mean; a direct analog of 'List Decoding' in the theory of error correcting codes. In this work, we develop an algorithm for list decodable mean estimation in the same setting achieving up to constants the information theoretically optimal recovery, optimal sample complexity, and in nearly linear time up to polylogarithmic factors in dimension. Our conceptual innovation is to design a descent style algorithm on a nonconvex landscape, iteratively removing minima to generate a succinct list of solutions. Our runtime bottleneck is a saddle-point optimization for which we design custom primal dual solvers for generalized packing and covering SDP's under Ky-Fan norms, which may be of independent interest.