Robust Estimators in High-Dimensions Without the Computational Intractability

Abstract

We study high-dimensional distribution learning in an agnostic setting where an adversary is allowed to arbitrarily corrupt an ϵ-fraction of the samples. Such questions have a rich history spanning statistics, machine learning, and theoretical computer science. Even in the most basic settings, the only known approaches are either computat ionally inefficient or lose dimensiondependent factors in their error guarantees. This raises the following question: Is high-dimensional agnostic distribution learning even possible, algorithmically? In this work, we obtain the first computationally efficient algorithms with dimension-independent error guarantees for agnostically learning several fundamental classes of high-dimensional distributions: (1) a single Gaussian, (2) a product distribution on the hypercube, (3) mixtures of two product distributions (under a natural balancedness condition), and (4) mixtures of spherical Gaussians. Our a lgorithms achieve error that is independent of the dimension, and in many cases scales nearly linearly with the fraction of adversarially corrupted samples. Moreover, we develop a general recipe for detecting and correcting corruptions in high-dimensions that may be applicable to many other problems. ©2019 Society for Industrial and Applied Mathematics.