Robust Estimators in High Dimensions without the Computational Intractability

TL;DR

This paper introduces the first efficient algorithms for high-dimensional distribution learning in adversarial settings, achieving dimension-independent error guarantees for various distribution classes.

Contribution

It presents novel computationally efficient algorithms with dimension-independent error bounds for learning several high-dimensional distributions under adversarial corruption.

Findings

Algorithms achieve error independent of dimension.

Error scales nearly-linearly with corrupted samples.

Applicable to multiple distribution classes.

Abstract

We study high-dimensional distribution learning in an agnostic setting where an adversary is allowed to arbitrarily corrupt an $\varepsilon$-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 computationally inefficient or lose dimension-dependent 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 algorithms 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.