# Robustly Learning a Gaussian: Getting Optimal Error, Efficiently

## TL;DR

This paper develops efficient algorithms for robustly estimating the parameters of high-dimensional Gaussian distributions with adversarial noise, achieving near-optimal error bounds with polynomial sample complexity.

## Contribution

It introduces robust estimators that attain optimal or near-optimal error bounds in high-dimensional Gaussian parameter estimation under adversarial noise, with polynomial runtime.

## Key findings

- Achieves estimation error $O(\varepsilon)$ in total variation distance.
- Provides polynomial-time algorithms for mean-only and mean-and-covariance estimation.
- Requires only polynomial samples, matching classical one-dimensional guarantees.

## Abstract

We study the fundamental problem of learning the parameters of a high-dimensional Gaussian in the presence of noise -- where an $\varepsilon$-fraction of our samples were chosen by an adversary. We give robust estimators that achieve estimation error $O(\varepsilon)$ in the total variation distance, which is optimal up to a universal constant that is independent of the dimension. In the case where just the mean is unknown, our robustness guarantee is optimal up to a factor of $\sqrt{2}$ and the running time is polynomial in $d$ and $1/\epsilon$. When both the mean and covariance are unknown, the running time is polynomial in $d$ and quasipolynomial in $1/\varepsilon$. Moreover all of our algorithms require only a polynomial number of samples. Our work shows that the same sorts of error guarantees that were established over fifty years ago in the one-dimensional setting can also be achieved by efficient algorithms in high-dimensional settings.